1348:
891:
1343:{\displaystyle {\begin{aligned}\sum _{r=0}^{m+n}{m+n \choose r}x^{r}&=(1+x)^{m+n}\\&=(1+x)^{m}(1+x)^{n}\\&={\biggl (}\sum _{i=0}^{m}{m \choose i}x^{i}{\biggr )}{\biggl (}\sum _{j=0}^{n}{n \choose j}x^{j}{\biggr )}\\&=\sum _{r=0}^{m+n}{\biggl (}\sum _{k=0}^{r}{m \choose k}{n \choose r-k}{\biggr )}x^{r},\end{aligned}}}
714:
2348:
455:
488:
3172:
2723:
177:
2127:
234:
3005:
872:
1586:
1711:
3370:
896:
2015:
2581:
709:{\displaystyle {\biggl (}\sum _{i=0}^{m}a_{i}x^{i}{\biggr )}{\biggl (}\sum _{j=0}^{n}b_{j}x^{j}{\biggr )}=\sum _{r=0}^{m+n}{\biggl (}\sum _{k=0}^{r}a_{k}b_{r-k}{\biggr )}x^{r},}
1855:
2445:
2416:
2387:
1471:
3258:
1780:
2533:
2511:
2489:
2467:
3210:
2872:
2827:
2788:
3032:
3390:
When both sides have been divided by the expression on the left, so that the sum is 1, then the terms of the sum may be interpreted as probabilities. The resulting
1353:
where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all
2343:{\displaystyle \sum _{k_{1}+\cdots +k_{p}=m}{n_{1} \choose k_{1}}{n_{2} \choose k_{2}}\cdots {n_{p} \choose k_{p}}={n_{1}+\dots +n_{p} \choose m}.}
450:{\displaystyle {n_{1}+\dots +n_{p} \choose m}=\sum _{k_{1}+\cdots +k_{p}=m}{n_{1} \choose k_{1}}{n_{2} \choose k_{2}}\cdots {n_{p} \choose k_{p}}.}
2605:
59:
3479:
2887:
759:
198:
2354:
1405:
3023:
2353:
This identity can be obtained through the algebraic derivation above when more than two polynomials are used, or through a simple
1498:
1716:
paths that start on the bottom left vertex and, moving only upwards or rightwards, end at the top right vertex (this is because
1618:
3278:
3484:
3395:
1951:
2538:
3433:
3379:
3391:
3213:
2092:) is confined to be within the square) to obtain the total number of paths that start at (0, 0) and end at (
219:
3461:, Regional Conference Series in Applied Mathematics, vol. 21, Philadelphia, PA: SIAM, pp. 59–60
3428:
1813:
2421:
2392:
2363:
1426:
3423:
3219:
1746:
50:
202:
2516:
2494:
2472:
2450:
3180:
3167:{\displaystyle \;_{2}F_{1}(a,b;c;1)={\frac {\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}}}
2832:
31:
1396:, both sides of Vandermonde's identity are zero due to the definition of binomial coefficients.
2793:
2754:
2878:
3015:
750:
3011:
2748:
3261:
2744:
2729:
228:
Vandermonde's identity can be generalized in numerous ways, including to the identity
3473:
3454:
3019:
2591:
The identity generalizes to non-integer arguments. In this case, it is known as the
38:
17:
2583:
in the left-hand side, which is also exactly what is done in the right-hand side.
1732:
up moves must be made (or vice versa) in any order, and the total path length is
3022:). The Chu–Vandermonde identity can also be seen to be a special case of
471:
205:
2718:{\displaystyle {s+t \choose n}=\sum _{k=0}^{n}{s \choose k}{t \choose n-k}}
172:{\displaystyle {m+n \choose r}=\sum _{k=0}^{r}{m \choose k}{n \choose r-k}}
212:
3398:. That is the probability distribution of the number of red marbles in
885:, and then the above formula for the product of polynomials, we obtain
183:
3000:{\displaystyle (s+t)_{n}=\sum _{k=0}^{n}{n \choose k}(s)_{k}(t)_{n-k}}
2049:
2743:. It can be proved along the lines of the algebraic proof above by
867:{\displaystyle (1+x)^{m+n}=\sum _{r=0}^{m+n}{m+n \choose r}x^{r}.}
1581:{\displaystyle \sum _{k=0}^{r}{m \choose k}{n \choose r-k}.}
3264:. One regains the Chu–Vandermonde identity by taking
3018:(for more on umbral variants of the binomial theorem, see
1706:{\displaystyle {\binom {r+(m+n-r)}{r}}={\binom {m+n}{r}}}
1913:) upward moves must be made and the path length must be
3365:{\displaystyle {n \choose k}=(-1)^{k}{k-n-1 \choose k}}
2877:
This identity may be rewritten in terms of the falling
1476:
The answer is also the sum over all possible values of
27:
Mathematical theorem on convolved binomial coefficients
2542:
2520:
2498:
2476:
2454:
2425:
2396:
2367:
2121:
3281:
3222:
3183:
3035:
2890:
2835:
2796:
2757:
2608:
2541:
2519:
2497:
2475:
2453:
2424:
2395:
2366:
2130:
1954:
1816:
1749:
1621:
1501:
1429:
894:
762:
491:
237:
201:(1772), although it was already known in 1303 by the
62:
1404:Vandermonde's identity also admits a combinatorial
3364:
3252:
3204:
3166:
2999:
2866:
2821:
2782:
2717:
2575:
2527:
2505:
2483:
2461:
2439:
2410:
2381:
2342:
2009:
1849:
1806:upward moves must be made (and the path length is
1774:
1705:
1580:
1465:
1376:, Vandermonde's identity follows for all integers
1342:
877:Using the binomial theorem also for the exponents
866:
708:
449:
171:
30:For the expression for a special determinant, see
3356:
3329:
3298:
3285:
2953:
2940:
2829:and comparing terms with the binomial series for
2709:
2688:
2679:
2666:
2633:
2612:
2331:
2290:
2278:
2251:
2239:
2212:
2203:
2176:
2001:
1980:
1971:
1958:
1841:
1820:
1766:
1753:
1697:
1676:
1664:
1625:
1569:
1548:
1539:
1526:
1454:
1433:
1318:
1310:
1289:
1280:
1267:
1238:
1194:
1176:
1163:
1134:
1127:
1109:
1096:
1067:
950:
929:
845:
824:
688:
634:
597:
549:
542:
494:
438:
411:
399:
372:
363:
336:
282:
241:
163:
142:
133:
120:
87:
66:
2010:{\displaystyle {\binom {m}{k}}{\binom {n}{r-k}}}
3010:in which form it is clearly recognizable as an
1480:, of the number of subcommittees consisting of
1416:women. In how many ways can a subcommittee of
1408:, as follows. Suppose a committee consists of
3382:is a further generalization of this identity.
2052:of all paths that start at (0, 0) and end at (
2576:{\displaystyle \textstyle n_{1}+\dots +n_{p}}
8:
3459:Orthogonal polynomials and special functions
3386:The hypergeometric probability distribution
3185:
3037:
3355:
3328:
3326:
3320:
3297:
3284:
3282:
3280:
3221:
3196:
3186:
3182:
3084:
3048:
3038:
3034:
2985:
2969:
2952:
2939:
2937:
2931:
2920:
2907:
2889:
2852:
2834:
2813:
2795:
2774:
2756:
2708:
2687:
2685:
2678:
2665:
2663:
2657:
2646:
2632:
2611:
2609:
2607:
2566:
2547:
2540:
2518:
2496:
2474:
2452:
2430:
2423:
2401:
2394:
2372:
2365:
2330:
2319:
2300:
2289:
2287:
2277:
2270:
2260:
2250:
2248:
2238:
2231:
2221:
2211:
2209:
2202:
2195:
2185:
2175:
2173:
2159:
2140:
2135:
2129:
2000:
1979:
1977:
1970:
1957:
1955:
1953:
1840:
1819:
1817:
1815:
1765:
1752:
1750:
1748:
1696:
1675:
1673:
1663:
1624:
1622:
1620:
1568:
1547:
1545:
1538:
1525:
1523:
1517:
1506:
1500:
1453:
1432:
1430:
1428:
1327:
1317:
1316:
1309:
1288:
1286:
1279:
1266:
1264:
1258:
1247:
1237:
1236:
1224:
1213:
1193:
1192:
1186:
1175:
1162:
1160:
1154:
1143:
1133:
1132:
1126:
1125:
1119:
1108:
1095:
1093:
1087:
1076:
1066:
1065:
1049:
1027:
989:
960:
949:
928:
926:
914:
903:
895:
893:
855:
844:
823:
821:
809:
798:
779:
761:
697:
687:
686:
674:
664:
654:
643:
633:
632:
620:
609:
596:
595:
589:
579:
569:
558:
548:
547:
541:
540:
534:
524:
514:
503:
493:
492:
490:
437:
430:
420:
410:
408:
398:
391:
381:
371:
369:
362:
355:
345:
335:
333:
319:
300:
295:
281:
270:
251:
240:
238:
236:
162:
141:
139:
132:
119:
117:
111:
100:
86:
65:
63:
61:
3445:
2447:out of another set, and so on, through
1740:). Call the bottom left vertex (0, 0).
1782:paths starting at (0, 0) that end at (
7:
2020:paths that start at (0, 0), end at (
2491:elements have been chosen from the
3333:
3289:
3223:
3143:
3125:
3099:
3087:
2944:
2692:
2670:
2616:
2294:
2255:
2216:
2180:
2117:Generalized Vandermonde's identity
1984:
1962:
1824:
1757:
1680:
1629:
1552:
1530:
1437:
1420:members be formed? The answer is
1293:
1271:
1167:
1100:
933:
828:
415:
376:
340:
245:
146:
124:
70:
25:
1850:{\displaystyle {\binom {n}{r-k}}}
719:where we use the convention that
2440:{\displaystyle \textstyle k_{2}}
2411:{\displaystyle \textstyle n_{1}}
2382:{\displaystyle \textstyle k_{1}}
1466:{\displaystyle {m+n \choose r}.}
741: = 0 for all integers
726: = 0 for all integers
49:) is the following identity for
3253:{\displaystyle \Gamma (n+1)=n!}
2389:elements out of a first set of
1775:{\displaystyle {\binom {m}{k}}}
470:In general, the product of two
199:Alexandre-Théophile Vandermonde
197:. The identity is named after
3317:
3307:
3238:
3226:
3158:
3146:
3140:
3128:
3120:
3102:
3096:
3090:
3078:
3054:
3024:Gauss's hypergeometric theorem
2982:
2975:
2966:
2959:
2904:
2891:
2849:
2836:
2810:
2797:
2771:
2758:
2596:
2593:Chu–Vandermonde identity
2587:Chu–Vandermonde identity
1655:
1637:
1046:
1033:
1024:
1011:
986:
973:
776:
763:
1:
3480:Factorial and binomial topics
2739:and any non-negative integer
2513:sets. One therefore chooses
2360:On the one hand, one chooses
1372:By comparing coefficients of
2528:{\displaystyle \textstyle m}
2506:{\displaystyle \textstyle p}
2484:{\displaystyle \textstyle m}
2469:such sets, until a total of
2462:{\displaystyle \textstyle p}
482:, respectively, is given by
3396:hypergeometric distribution
3205:{\displaystyle \;_{2}F_{1}}
2867:{\displaystyle (1+x)^{s+t}}
2597:Askey 1975, pp. 59–60
1596:Take a rectangular grid of
218:to this theorem called the
3501:
3272:and applying the identity
29:
2822:{\displaystyle (1+x)^{t}}
2783:{\displaystyle (1+x)^{s}}
47:Vandermonde's convolution
3392:probability distribution
2116:
1810:). Similarly, there are
3406:from an urn containing
3214:hypergeometric function
3366:
3254:
3206:
3168:
3001:
2936:
2868:
2823:
2784:
2719:
2662:
2577:
2529:
2507:
2485:
2463:
2441:
2412:
2383:
2344:
2011:
1851:
1776:
1707:
1582:
1522:
1467:
1392:. For larger integers
1344:
1263:
1235:
1159:
1092:
925:
868:
820:
710:
659:
631:
574:
519:
451:
173:
116:
43:Vandermonde's identity
3429:Hockey-stick identity
3367:
3255:
3207:
3169:
3002:
2916:
2869:
2824:
2785:
2720:
2642:
2599:) and takes the form
2578:
2530:
2508:
2486:
2464:
2442:
2413:
2384:
2345:
2012:
1852:
1777:
1708:
1612:) squares. There are
1583:
1502:
1468:
1406:double counting proof
1345:
1243:
1209:
1139:
1072:
899:
869:
794:
711:
639:
605:
554:
499:
452:
223:-Vandermonde identity
203:Chinese mathematician
174:
96:
51:binomial coefficients
3485:Algebraic identities
3434:Rothe–Hagen identity
3380:Rothe–Hagen identity
3279:
3220:
3181:
3033:
3026:, which states that
2888:
2833:
2794:
2755:
2606:
2539:
2517:
2495:
2473:
2451:
2422:
2393:
2364:
2128:
1952:
1814:
1747:
1619:
1499:
1427:
892:
760:
489:
235:
182:for any nonnegative
60:
18:Vandermonde identity
3404:without replacement
2036:), and go through (
1857:paths starting at (
1400:Combinatorial proof
1380:with 0 ≤
3362:
3250:
3202:
3164:
2997:
2879:Pochhammer symbols
2864:
2819:
2780:
2715:
2573:
2572:
2525:
2524:
2503:
2502:
2481:
2480:
2459:
2458:
2437:
2436:
2408:
2407:
2379:
2378:
2340:
2172:
2007:
1847:
1772:
1703:
1578:
1463:
1340:
1338:
864:
706:
447:
332:
169:
32:Vandermonde matrix
3424:Pascal's identity
3354:
3296:
3162:
2951:
2707:
2677:
2631:
2329:
2276:
2237:
2201:
2131:
1999:
1969:
1945:. Thus there are
1893:right moves and (
1885:), as a total of
1839:
1764:
1695:
1662:
1592:Geometrical proof
1567:
1537:
1452:
1308:
1278:
1174:
1107:
948:
843:
436:
397:
361:
291:
280:
161:
131:
85:
16:(Redirected from
3492:
3464:
3463:for the history.
3462:
3450:
3371:
3369:
3368:
3363:
3361:
3360:
3359:
3350:
3332:
3325:
3324:
3303:
3302:
3301:
3288:
3259:
3257:
3256:
3251:
3211:
3209:
3208:
3203:
3201:
3200:
3191:
3190:
3173:
3171:
3170:
3165:
3163:
3161:
3123:
3085:
3053:
3052:
3043:
3042:
3016:binomial theorem
3006:
3004:
3003:
2998:
2996:
2995:
2974:
2973:
2958:
2957:
2956:
2943:
2935:
2930:
2912:
2911:
2873:
2871:
2870:
2865:
2863:
2862:
2828:
2826:
2825:
2820:
2818:
2817:
2789:
2787:
2786:
2781:
2779:
2778:
2724:
2722:
2721:
2716:
2714:
2713:
2712:
2706:
2691:
2684:
2683:
2682:
2669:
2661:
2656:
2638:
2637:
2636:
2627:
2615:
2582:
2580:
2579:
2574:
2571:
2570:
2552:
2551:
2535:elements out of
2534:
2532:
2531:
2526:
2512:
2510:
2509:
2504:
2490:
2488:
2487:
2482:
2468:
2466:
2465:
2460:
2446:
2444:
2443:
2438:
2435:
2434:
2417:
2415:
2414:
2409:
2406:
2405:
2388:
2386:
2385:
2380:
2377:
2376:
2349:
2347:
2346:
2341:
2336:
2335:
2334:
2325:
2324:
2323:
2305:
2304:
2293:
2283:
2282:
2281:
2275:
2274:
2265:
2264:
2254:
2244:
2243:
2242:
2236:
2235:
2226:
2225:
2215:
2208:
2207:
2206:
2200:
2199:
2190:
2189:
2179:
2171:
2164:
2163:
2145:
2144:
2016:
2014:
2013:
2008:
2006:
2005:
2004:
1998:
1983:
1976:
1975:
1974:
1961:
1856:
1854:
1853:
1848:
1846:
1845:
1844:
1838:
1823:
1798:right moves and
1781:
1779:
1778:
1773:
1771:
1770:
1769:
1756:
1720:right moves and
1712:
1710:
1709:
1704:
1702:
1701:
1700:
1691:
1679:
1669:
1668:
1667:
1658:
1628:
1587:
1585:
1584:
1579:
1574:
1573:
1572:
1566:
1551:
1544:
1543:
1542:
1529:
1521:
1516:
1472:
1470:
1469:
1464:
1459:
1458:
1457:
1448:
1436:
1369:, respectively.
1365: >
1357: >
1349:
1347:
1346:
1341:
1339:
1332:
1331:
1322:
1321:
1315:
1314:
1313:
1307:
1292:
1285:
1284:
1283:
1270:
1262:
1257:
1242:
1241:
1234:
1223:
1202:
1198:
1197:
1191:
1190:
1181:
1180:
1179:
1166:
1158:
1153:
1138:
1137:
1131:
1130:
1124:
1123:
1114:
1113:
1112:
1099:
1091:
1086:
1071:
1070:
1058:
1054:
1053:
1032:
1031:
1004:
1000:
999:
965:
964:
955:
954:
953:
944:
932:
924:
913:
873:
871:
870:
865:
860:
859:
850:
849:
848:
839:
827:
819:
808:
790:
789:
751:binomial theorem
745: >
730: >
715:
713:
712:
707:
702:
701:
692:
691:
685:
684:
669:
668:
658:
653:
638:
637:
630:
619:
601:
600:
594:
593:
584:
583:
573:
568:
553:
552:
546:
545:
539:
538:
529:
528:
518:
513:
498:
497:
456:
454:
453:
448:
443:
442:
441:
435:
434:
425:
424:
414:
404:
403:
402:
396:
395:
386:
385:
375:
368:
367:
366:
360:
359:
350:
349:
339:
331:
324:
323:
305:
304:
287:
286:
285:
276:
275:
274:
256:
255:
244:
178:
176:
175:
170:
168:
167:
166:
160:
145:
138:
137:
136:
123:
115:
110:
92:
91:
90:
81:
69:
21:
3500:
3499:
3495:
3494:
3493:
3491:
3490:
3489:
3470:
3469:
3468:
3467:
3453:
3451:
3447:
3442:
3420:
3388:
3334:
3327:
3316:
3283:
3277:
3276:
3218:
3217:
3192:
3184:
3179:
3178:
3124:
3086:
3044:
3036:
3031:
3030:
3014:variant of the
2981:
2965:
2938:
2903:
2886:
2885:
2848:
2831:
2830:
2809:
2792:
2791:
2770:
2753:
2752:
2749:binomial series
2696:
2686:
2664:
2617:
2610:
2604:
2603:
2589:
2562:
2543:
2537:
2536:
2515:
2514:
2493:
2492:
2471:
2470:
2449:
2448:
2426:
2420:
2419:
2418:elements; then
2397:
2391:
2390:
2368:
2362:
2361:
2355:double counting
2315:
2296:
2295:
2288:
2266:
2256:
2249:
2227:
2217:
2210:
2191:
2181:
2174:
2155:
2136:
2126:
2125:
2119:
2114:
2112:Generalizations
2080:(as the point (
2068:), so sum from
1988:
1978:
1956:
1950:
1949:
1869:) that end at (
1828:
1818:
1812:
1811:
1751:
1745:
1744:
1736: +
1681:
1674:
1630:
1623:
1617:
1616:
1594:
1556:
1546:
1524:
1497:
1496:
1488: −
1438:
1431:
1425:
1424:
1402:
1388: +
1337:
1336:
1323:
1297:
1287:
1265:
1200:
1199:
1182:
1161:
1115:
1094:
1056:
1055:
1045:
1023:
1002:
1001:
985:
966:
956:
934:
927:
890:
889:
851:
829:
822:
775:
758:
757:
739:
724:
693:
670:
660:
585:
575:
530:
520:
487:
486:
468:
466:Algebraic proof
463:
426:
416:
409:
387:
377:
370:
351:
341:
334:
315:
296:
266:
247:
246:
239:
233:
232:
150:
140:
118:
71:
64:
58:
57:
35:
28:
23:
22:
15:
12:
11:
5:
3498:
3496:
3488:
3487:
3482:
3472:
3471:
3466:
3465:
3455:Askey, Richard
3444:
3443:
3441:
3438:
3437:
3436:
3431:
3426:
3419:
3416:
3414:blue marbles.
3387:
3384:
3373:
3372:
3358:
3353:
3349:
3346:
3343:
3340:
3337:
3331:
3323:
3319:
3315:
3312:
3309:
3306:
3300:
3295:
3292:
3287:
3268: = −
3262:gamma function
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3199:
3195:
3189:
3175:
3174:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3051:
3047:
3041:
3008:
3007:
2994:
2991:
2988:
2984:
2980:
2977:
2972:
2968:
2964:
2961:
2955:
2950:
2947:
2942:
2934:
2929:
2926:
2923:
2919:
2915:
2910:
2906:
2902:
2899:
2896:
2893:
2861:
2858:
2855:
2851:
2847:
2844:
2841:
2838:
2816:
2812:
2808:
2805:
2802:
2799:
2777:
2773:
2769:
2766:
2763:
2760:
2730:complex-valued
2726:
2725:
2711:
2705:
2702:
2699:
2695:
2690:
2681:
2676:
2673:
2668:
2660:
2655:
2652:
2649:
2645:
2641:
2635:
2630:
2626:
2623:
2620:
2614:
2588:
2585:
2569:
2565:
2561:
2558:
2555:
2550:
2546:
2523:
2501:
2479:
2457:
2433:
2429:
2404:
2400:
2375:
2371:
2351:
2350:
2339:
2333:
2328:
2322:
2318:
2314:
2311:
2308:
2303:
2299:
2292:
2286:
2280:
2273:
2269:
2263:
2259:
2253:
2247:
2241:
2234:
2230:
2224:
2220:
2214:
2205:
2198:
2194:
2188:
2184:
2178:
2170:
2167:
2162:
2158:
2154:
2151:
2148:
2143:
2139:
2134:
2118:
2115:
2113:
2110:
2018:
2017:
2003:
1997:
1994:
1991:
1987:
1982:
1973:
1968:
1965:
1960:
1843:
1837:
1834:
1831:
1827:
1822:
1768:
1763:
1760:
1755:
1714:
1713:
1699:
1694:
1690:
1687:
1684:
1678:
1672:
1666:
1661:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1627:
1593:
1590:
1589:
1588:
1577:
1571:
1565:
1562:
1559:
1555:
1550:
1541:
1536:
1533:
1528:
1520:
1515:
1512:
1509:
1505:
1474:
1473:
1462:
1456:
1451:
1447:
1444:
1441:
1435:
1401:
1398:
1351:
1350:
1335:
1330:
1326:
1320:
1312:
1306:
1303:
1300:
1296:
1291:
1282:
1277:
1274:
1269:
1261:
1256:
1253:
1250:
1246:
1240:
1233:
1230:
1227:
1222:
1219:
1216:
1212:
1208:
1205:
1203:
1201:
1196:
1189:
1185:
1178:
1173:
1170:
1165:
1157:
1152:
1149:
1146:
1142:
1136:
1129:
1122:
1118:
1111:
1106:
1103:
1098:
1090:
1085:
1082:
1079:
1075:
1069:
1064:
1061:
1059:
1057:
1052:
1048:
1044:
1041:
1038:
1035:
1030:
1026:
1022:
1019:
1016:
1013:
1010:
1007:
1005:
1003:
998:
995:
992:
988:
984:
981:
978:
975:
972:
969:
967:
963:
959:
952:
947:
943:
940:
937:
931:
923:
920:
917:
912:
909:
906:
902:
898:
897:
875:
874:
863:
858:
854:
847:
842:
838:
835:
832:
826:
818:
815:
812:
807:
804:
801:
797:
793:
788:
785:
782:
778:
774:
771:
768:
765:
737:
722:
717:
716:
705:
700:
696:
690:
683:
680:
677:
673:
667:
663:
657:
652:
649:
646:
642:
636:
629:
626:
623:
618:
615:
612:
608:
604:
599:
592:
588:
582:
578:
572:
567:
564:
561:
557:
551:
544:
537:
533:
527:
523:
517:
512:
509:
506:
502:
496:
467:
464:
462:
459:
458:
457:
446:
440:
433:
429:
423:
419:
413:
407:
401:
394:
390:
384:
380:
374:
365:
358:
354:
348:
344:
338:
330:
327:
322:
318:
314:
311:
308:
303:
299:
294:
290:
284:
279:
273:
269:
265:
262:
259:
254:
250:
243:
180:
179:
165:
159:
156:
153:
149:
144:
135:
130:
127:
122:
114:
109:
106:
103:
99:
95:
89:
84:
80:
77:
74:
68:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3497:
3486:
3483:
3481:
3478:
3477:
3475:
3460:
3456:
3449:
3446:
3439:
3435:
3432:
3430:
3427:
3425:
3422:
3421:
3417:
3415:
3413:
3409:
3405:
3401:
3397:
3393:
3385:
3383:
3381:
3376:
3351:
3347:
3344:
3341:
3338:
3335:
3321:
3313:
3310:
3304:
3293:
3290:
3275:
3274:
3273:
3271:
3267:
3263:
3247:
3244:
3241:
3235:
3232:
3229:
3215:
3197:
3193:
3187:
3155:
3152:
3149:
3137:
3134:
3131:
3117:
3114:
3111:
3108:
3105:
3093:
3081:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3049:
3045:
3039:
3029:
3028:
3027:
3025:
3021:
3020:binomial type
3017:
3013:
2992:
2989:
2986:
2978:
2970:
2962:
2948:
2945:
2932:
2927:
2924:
2921:
2917:
2913:
2908:
2900:
2897:
2894:
2884:
2883:
2882:
2880:
2875:
2859:
2856:
2853:
2845:
2842:
2839:
2814:
2806:
2803:
2800:
2775:
2767:
2764:
2761:
2750:
2746:
2742:
2738:
2734:
2731:
2703:
2700:
2697:
2693:
2674:
2671:
2658:
2653:
2650:
2647:
2643:
2639:
2628:
2624:
2621:
2618:
2602:
2601:
2600:
2598:
2594:
2586:
2584:
2567:
2563:
2559:
2556:
2553:
2548:
2544:
2521:
2499:
2477:
2455:
2431:
2427:
2402:
2398:
2373:
2369:
2358:
2356:
2337:
2326:
2320:
2316:
2312:
2309:
2306:
2301:
2297:
2284:
2271:
2267:
2261:
2257:
2245:
2232:
2228:
2222:
2218:
2196:
2192:
2186:
2182:
2168:
2165:
2160:
2156:
2152:
2149:
2146:
2141:
2137:
2132:
2124:
2123:
2122:
2111:
2109:
2107:
2103:
2099:
2095:
2091:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2059:
2055:
2051:
2048:). This is a
2047:
2043:
2039:
2035:
2031:
2027:
2023:
1995:
1992:
1989:
1985:
1966:
1963:
1948:
1947:
1946:
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1912:
1908:
1904:
1900:
1896:
1892:
1888:
1884:
1880:
1876:
1872:
1868:
1864:
1860:
1835:
1832:
1829:
1825:
1809:
1805:
1801:
1797:
1793:
1789:
1785:
1761:
1758:
1741:
1739:
1735:
1731:
1727:
1723:
1719:
1692:
1688:
1685:
1682:
1670:
1659:
1652:
1649:
1646:
1643:
1640:
1634:
1631:
1615:
1614:
1613:
1611:
1607:
1603:
1599:
1591:
1575:
1563:
1560:
1557:
1553:
1534:
1531:
1518:
1513:
1510:
1507:
1503:
1495:
1494:
1493:
1491:
1487:
1483:
1479:
1460:
1449:
1445:
1442:
1439:
1423:
1422:
1421:
1419:
1415:
1411:
1407:
1399:
1397:
1395:
1391:
1387:
1384: ≤
1383:
1379:
1375:
1370:
1368:
1364:
1360:
1356:
1333:
1328:
1324:
1304:
1301:
1298:
1294:
1275:
1272:
1259:
1254:
1251:
1248:
1244:
1231:
1228:
1225:
1220:
1217:
1214:
1210:
1206:
1204:
1187:
1183:
1171:
1168:
1155:
1150:
1147:
1144:
1140:
1120:
1116:
1104:
1101:
1088:
1083:
1080:
1077:
1073:
1062:
1060:
1050:
1042:
1039:
1036:
1028:
1020:
1017:
1014:
1008:
1006:
996:
993:
990:
982:
979:
976:
970:
968:
961:
957:
945:
941:
938:
935:
921:
918:
915:
910:
907:
904:
900:
888:
887:
886:
884:
880:
861:
856:
852:
840:
836:
833:
830:
816:
813:
810:
805:
802:
799:
795:
791:
786:
783:
780:
772:
769:
766:
756:
755:
754:
752:
748:
744:
740:
733:
729:
725:
703:
698:
694:
681:
678:
675:
671:
665:
661:
655:
650:
647:
644:
640:
627:
624:
621:
616:
613:
610:
606:
602:
590:
586:
580:
576:
570:
565:
562:
559:
555:
535:
531:
525:
521:
515:
510:
507:
504:
500:
485:
484:
483:
481:
477:
474:with degrees
473:
465:
460:
444:
431:
427:
421:
417:
405:
392:
388:
382:
378:
356:
352:
346:
342:
328:
325:
320:
316:
312:
309:
306:
301:
297:
292:
288:
277:
271:
267:
263:
260:
257:
252:
248:
231:
230:
229:
226:
224:
222:
217:
215:
209:
207:
204:
200:
196:
192:
188:
185:
157:
154:
151:
147:
128:
125:
112:
107:
104:
101:
97:
93:
82:
78:
75:
72:
56:
55:
54:
52:
48:
44:
40:
39:combinatorics
33:
19:
3458:
3448:
3411:
3407:
3403:
3399:
3389:
3377:
3374:
3269:
3265:
3176:
3009:
2876:
2740:
2736:
2732:
2728:for general
2727:
2592:
2590:
2359:
2352:
2120:
2105:
2101:
2097:
2093:
2089:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2045:
2041:
2037:
2033:
2029:
2025:
2021:
2019:
1942:
1938:
1934:
1930:
1926:
1922:
1918:
1914:
1910:
1906:
1902:
1898:
1894:
1890:
1886:
1882:
1878:
1874:
1870:
1866:
1862:
1858:
1807:
1803:
1799:
1795:
1791:
1787:
1783:
1742:
1737:
1733:
1729:
1725:
1721:
1717:
1715:
1609:
1605:
1601:
1597:
1595:
1489:
1485:
1481:
1477:
1475:
1417:
1413:
1409:
1403:
1393:
1389:
1385:
1381:
1377:
1373:
1371:
1366:
1362:
1358:
1354:
1352:
882:
878:
876:
746:
742:
735:
731:
727:
720:
718:
479:
475:
469:
227:
220:
213:
210:
194:
190:
186:
181:
46:
42:
36:
3375:liberally.
2745:multiplying
472:polynomials
211:There is a
3474:Categories
3440:References
2357:argument.
1743:There are
206:Zhu Shijie
3345:−
3339:−
3311:−
3224:Γ
3153:−
3144:Γ
3135:−
3126:Γ
3115:−
3109:−
3100:Γ
3088:Γ
2990:−
2918:∑
2701:−
2644:∑
2557:⋯
2310:⋯
2246:⋯
2150:⋯
2133:∑
1993:−
1833:−
1650:−
1561:−
1504:∑
1302:−
1245:∑
1211:∑
1141:∑
1074:∑
901:∑
796:∑
749:. By the
679:−
641:∑
607:∑
556:∑
501:∑
406:⋯
310:⋯
293:∑
261:⋯
155:−
98:∑
3457:(1975),
3418:See also
3410:red and
1484:men and
1412:men and
184:integers
3394:is the
3260:is the
3212:is the
2072:= 0 to
1492:women:
216:-analog
3402:draws
3177:where
3012:umbral
2050:subset
1794:), as
461:Proofs
2595:(see
1933:) − (
1905:) − (
3452:See
3378:The
3216:and
2790:and
2751:for
2747:the
2735:and
1941:) =
1361:and
881:and
734:and
478:and
45:(or
2881:as
2108:).
1921:+ (
1600:x (
37:In
3476::
2874:.
2096:,
2084:,
2076:=
2056:,
2040:,
2024:,
1873:,
1861:,
1786:,
753:,
225:.
208:.
193:,
189:,
53::
41:,
3412:m
3408:n
3400:r
3357:)
3352:k
3348:1
3342:n
3336:k
3330:(
3322:k
3318:)
3314:1
3308:(
3305:=
3299:)
3294:k
3291:n
3286:(
3270:n
3266:a
3248:!
3245:n
3242:=
3239:)
3236:1
3233:+
3230:n
3227:(
3198:1
3194:F
3188:2
3159:)
3156:b
3150:c
3147:(
3141:)
3138:a
3132:c
3129:(
3121:)
3118:b
3112:a
3106:c
3103:(
3097:)
3094:c
3091:(
3082:=
3079:)
3076:1
3073:;
3070:c
3067:;
3064:b
3061:,
3058:a
3055:(
3050:1
3046:F
3040:2
2993:k
2987:n
2983:)
2979:t
2976:(
2971:k
2967:)
2963:s
2960:(
2954:)
2949:k
2946:n
2941:(
2933:n
2928:0
2925:=
2922:k
2914:=
2909:n
2905:)
2901:t
2898:+
2895:s
2892:(
2860:t
2857:+
2854:s
2850:)
2846:x
2843:+
2840:1
2837:(
2815:t
2811:)
2807:x
2804:+
2801:1
2798:(
2776:s
2772:)
2768:x
2765:+
2762:1
2759:(
2741:n
2737:t
2733:s
2710:)
2704:k
2698:n
2694:t
2689:(
2680:)
2675:k
2672:s
2667:(
2659:n
2654:0
2651:=
2648:k
2640:=
2634:)
2629:n
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2622:+
2619:s
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2568:p
2564:n
2560:+
2554:+
2549:1
2545:n
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2500:p
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2428:k
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2338:.
2332:)
2327:m
2321:p
2317:n
2313:+
2307:+
2302:1
2298:n
2291:(
2285:=
2279:)
2272:p
2268:k
2262:p
2258:n
2252:(
2240:)
2233:2
2229:k
2223:2
2219:n
2213:(
2204:)
2197:1
2193:k
2187:1
2183:n
2177:(
2169:m
2166:=
2161:p
2157:k
2153:+
2147:+
2142:1
2138:k
2106:r
2104:−
2102:n
2100:+
2098:m
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2088:−
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2082:k
2078:r
2074:k
2070:k
2066:r
2064:−
2062:n
2060:+
2058:m
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2046:k
2044:−
2042:m
2038:k
2034:r
2032:−
2030:n
2028:+
2026:m
2022:r
2002:)
1996:k
1990:r
1986:n
1981:(
1972:)
1967:k
1964:m
1959:(
1943:n
1939:k
1937:−
1935:m
1931:r
1929:−
1927:n
1925:+
1923:m
1919:k
1917:−
1915:r
1911:k
1909:−
1907:m
1903:r
1901:−
1899:n
1897:+
1895:m
1891:k
1889:−
1887:r
1883:r
1881:−
1879:n
1877:+
1875:m
1871:r
1867:k
1865:−
1863:m
1859:k
1842:)
1836:k
1830:r
1826:n
1821:(
1808:m
1804:k
1802:−
1800:m
1796:k
1792:k
1790:−
1788:m
1784:k
1767:)
1762:k
1759:m
1754:(
1738:n
1734:m
1730:r
1728:-
1726:n
1724:+
1722:m
1718:r
1698:)
1693:r
1689:n
1686:+
1683:m
1677:(
1671:=
1665:)
1660:r
1656:)
1653:r
1647:n
1644:+
1641:m
1638:(
1635:+
1632:r
1626:(
1610:r
1608:−
1606:n
1604:+
1602:m
1598:r
1576:.
1570:)
1564:k
1558:r
1554:n
1549:(
1540:)
1535:k
1532:m
1527:(
1519:r
1514:0
1511:=
1508:k
1490:k
1486:r
1482:k
1478:k
1461:.
1455:)
1450:r
1446:n
1443:+
1440:m
1434:(
1418:r
1414:n
1410:m
1394:r
1390:n
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1382:r
1378:r
1374:x
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1355:i
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1329:r
1325:x
1319:)
1311:)
1305:k
1299:r
1295:n
1290:(
1281:)
1276:k
1273:m
1268:(
1260:r
1255:0
1252:=
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1239:(
1232:n
1229:+
1226:m
1221:0
1218:=
1215:r
1207:=
1195:)
1188:j
1184:x
1177:)
1172:j
1169:n
1164:(
1156:n
1151:0
1148:=
1145:j
1135:(
1128:)
1121:i
1117:x
1110:)
1105:i
1102:m
1097:(
1089:m
1084:0
1081:=
1078:i
1068:(
1063:=
1051:n
1047:)
1043:x
1040:+
1037:1
1034:(
1029:m
1025:)
1021:x
1018:+
1015:1
1012:(
1009:=
997:n
994:+
991:m
987:)
983:x
980:+
977:1
974:(
971:=
962:r
958:x
951:)
946:r
942:n
939:+
936:m
930:(
922:n
919:+
916:m
911:0
908:=
905:r
883:n
879:m
862:.
857:r
853:x
846:)
841:r
837:n
834:+
831:m
825:(
817:n
814:+
811:m
806:0
803:=
800:r
792:=
787:n
784:+
781:m
777:)
773:x
770:+
767:1
764:(
747:n
743:j
738:j
736:b
732:m
728:i
723:i
721:a
704:,
699:r
695:x
689:)
682:k
676:r
672:b
666:k
662:a
656:r
651:0
648:=
645:k
635:(
628:n
625:+
622:m
617:0
614:=
611:r
603:=
598:)
591:j
587:x
581:j
577:b
571:n
566:0
563:=
560:j
550:(
543:)
536:i
532:x
526:i
522:a
516:m
511:0
508:=
505:i
495:(
480:n
476:m
445:.
439:)
432:p
428:k
422:p
418:n
412:(
400:)
393:2
389:k
383:2
379:n
373:(
364:)
357:1
353:k
347:1
343:n
337:(
329:m
326:=
321:p
317:k
313:+
307:+
302:1
298:k
289:=
283:)
278:m
272:p
268:n
264:+
258:+
253:1
249:n
242:(
221:q
214:q
195:n
191:m
187:r
164:)
158:k
152:r
148:n
143:(
134:)
129:k
126:m
121:(
113:r
108:0
105:=
102:k
94:=
88:)
83:r
79:n
76:+
73:m
67:(
34:.
20:)
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