1337:
880:
1332:{\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}}}
703:
2337:
444:
477:
3161:
2712:
166:
2116:
223:
2994:
861:
1575:
1700:
3359:
885:
2004:
2570:
698:{\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},}
1844:
2434:
2405:
2376:
1460:
3247:
1769:
2522:
2500:
2478:
2456:
3199:
2861:
2816:
2777:
3021:
3379:
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
1342:
where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all
2332:{\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}.}
439:{\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}}.}
2594:
48:
3468:
2876:
748:
187:
2343:
1394:
3012:
2342:
This identity can be obtained through the algebraic derivation above when more than two polynomials are used, or through a simple
1487:
1705:
paths that start on the bottom left vertex and, moving only upwards or rightwards, end at the top right vertex (this is because
1607:
3267:
3473:
3384:
1940:
2527:
3422:
3368:
3380:
3202:
2081:) is confined to be within the square) to obtain the total number of paths that start at (0, 0) and end at (
208:
3450:, Regional Conference Series in Applied Mathematics, vol. 21, Philadelphia, PA: SIAM, pp. 59–60
3417:
1802:
2410:
2381:
2352:
1415:
3412:
3208:
1735:
39:
191:
2505:
2483:
2461:
2439:
3169:
3156:{\displaystyle \;_{2}F_{1}(a,b;c;1)={\frac {\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}}}
2821:
20:
1385:, both sides of Vandermonde's identity are zero due to the definition of binomial coefficients.
2782:
2743:
2867:
3004:
739:
3000:
2737:
3250:
2733:
2718:
217:
Vandermonde's identity can be generalized in numerous ways, including to the identity
3462:
3443:
3008:
2580:
The identity generalizes to non-integer arguments. In this case, it is known as the
27:
2572:
in the left-hand side, which is also exactly what is done in the right-hand side.
1721:
up moves must be made (or vice versa) in any order, and the total path length is
3011:). The Chu–Vandermonde identity can also be seen to be a special case of
460:
194:
2707:{\displaystyle {s+t \choose n}=\sum _{k=0}^{n}{s \choose k}{t \choose n-k}}
161:{\displaystyle {m+n \choose r}=\sum _{k=0}^{r}{m \choose k}{n \choose r-k}}
201:
3387:. That is the probability distribution of the number of red marbles in
874:, and then the above formula for the product of polynomials, we obtain
172:
2989:{\displaystyle (s+t)_{n}=\sum _{k=0}^{n}{n \choose k}(s)_{k}(t)_{n-k}}
2038:
2732:. It can be proved along the lines of the algebraic proof above by
856:{\displaystyle (1+x)^{m+n}=\sum _{r=0}^{m+n}{m+n \choose r}x^{r}.}
1570:{\displaystyle \sum _{k=0}^{r}{m \choose k}{n \choose r-k}.}
3253:. One regains the Chu–Vandermonde identity by taking
3007:(for more on umbral variants of the binomial theorem, see
1695:{\displaystyle {\binom {r+(m+n-r)}{r}}={\binom {m+n}{r}}}
1902:) upward moves must be made and the path length must be
3354:{\displaystyle {n \choose k}=(-1)^{k}{k-n-1 \choose k}}
2866:
This identity may be rewritten in terms of the falling
1465:
The answer is also the sum over all possible values of
16:
Mathematical theorem on convolved binomial coefficients
2531:
2509:
2487:
2465:
2443:
2414:
2385:
2356:
2110:
3270:
3211:
3172:
3024:
2879:
2824:
2785:
2746:
2597:
2530:
2508:
2486:
2464:
2442:
2413:
2384:
2355:
2119:
1943:
1805:
1738:
1610:
1490:
1418:
883:
751:
480:
226:
190:(1772), although it was already known in 1303 by the
51:
1393:Vandermonde's identity also admits a combinatorial
3353:
3241:
3193:
3155:
2988:
2855:
2810:
2771:
2706:
2564:
2516:
2494:
2472:
2450:
2428:
2399:
2370:
2331:
1998:
1838:
1795:upward moves must be made (and the path length is
1763:
1694:
1569:
1454:
1365:, Vandermonde's identity follows for all integers
1331:
866:Using the binomial theorem also for the exponents
855:
697:
438:
160:
19:For the expression for a special determinant, see
3345:
3318:
3287:
3274:
2942:
2929:
2818:and comparing terms with the binomial series for
2698:
2677:
2668:
2655:
2622:
2601:
2320:
2279:
2267:
2240:
2228:
2201:
2192:
2165:
1990:
1969:
1960:
1947:
1830:
1809:
1755:
1742:
1686:
1665:
1653:
1614:
1558:
1537:
1528:
1515:
1443:
1422:
1307:
1299:
1278:
1269:
1256:
1227:
1183:
1165:
1152:
1123:
1116:
1098:
1085:
1056:
939:
918:
834:
813:
677:
623:
586:
538:
531:
483:
427:
400:
388:
361:
352:
325:
271:
230:
152:
131:
122:
109:
76:
55:
1999:{\displaystyle {\binom {m}{k}}{\binom {n}{r-k}}}
2999:in which form it is clearly recognizable as an
1469:, of the number of subcommittees consisting of
1405:women. In how many ways can a subcommittee of
1397:, as follows. Suppose a committee consists of
3371:is a further generalization of this identity.
2041:of all paths that start at (0, 0) and end at (
2565:{\displaystyle \textstyle n_{1}+\dots +n_{p}}
8:
3448:Orthogonal polynomials and special functions
3375:The hypergeometric probability distribution
3174:
3026:
3344:
3317:
3315:
3309:
3286:
3273:
3271:
3269:
3210:
3185:
3175:
3171:
3073:
3037:
3027:
3023:
2974:
2958:
2941:
2928:
2926:
2920:
2909:
2896:
2878:
2841:
2823:
2802:
2784:
2763:
2745:
2697:
2676:
2674:
2667:
2654:
2652:
2646:
2635:
2621:
2600:
2598:
2596:
2555:
2536:
2529:
2507:
2485:
2463:
2441:
2419:
2412:
2390:
2383:
2361:
2354:
2319:
2308:
2289:
2278:
2276:
2266:
2259:
2249:
2239:
2237:
2227:
2220:
2210:
2200:
2198:
2191:
2184:
2174:
2164:
2162:
2148:
2129:
2124:
2118:
1989:
1968:
1966:
1959:
1946:
1944:
1942:
1829:
1808:
1806:
1804:
1754:
1741:
1739:
1737:
1685:
1664:
1662:
1652:
1613:
1611:
1609:
1557:
1536:
1534:
1527:
1514:
1512:
1506:
1495:
1489:
1442:
1421:
1419:
1417:
1316:
1306:
1305:
1298:
1277:
1275:
1268:
1255:
1253:
1247:
1236:
1226:
1225:
1213:
1202:
1182:
1181:
1175:
1164:
1151:
1149:
1143:
1132:
1122:
1121:
1115:
1114:
1108:
1097:
1084:
1082:
1076:
1065:
1055:
1054:
1038:
1016:
978:
949:
938:
917:
915:
903:
892:
884:
882:
844:
833:
812:
810:
798:
787:
768:
750:
686:
676:
675:
663:
653:
643:
632:
622:
621:
609:
598:
585:
584:
578:
568:
558:
547:
537:
536:
530:
529:
523:
513:
503:
492:
482:
481:
479:
426:
419:
409:
399:
397:
387:
380:
370:
360:
358:
351:
344:
334:
324:
322:
308:
289:
284:
270:
259:
240:
229:
227:
225:
151:
130:
128:
121:
108:
106:
100:
89:
75:
54:
52:
50:
3434:
2436:out of another set, and so on, through
1729:). Call the bottom left vertex (0, 0).
1771:paths starting at (0, 0) that end at (
7:
2009:paths that start at (0, 0), end at (
2480:elements have been chosen from the
3322:
3278:
3212:
3132:
3114:
3088:
3076:
2933:
2681:
2659:
2605:
2283:
2244:
2205:
2169:
2106:Generalized Vandermonde's identity
1973:
1951:
1813:
1746:
1669:
1618:
1541:
1519:
1426:
1409:members be formed? The answer is
1282:
1260:
1156:
1089:
922:
817:
404:
365:
329:
234:
135:
113:
59:
14:
1839:{\displaystyle {\binom {n}{r-k}}}
708:where we use the convention that
2429:{\displaystyle \textstyle k_{2}}
2400:{\displaystyle \textstyle n_{1}}
2371:{\displaystyle \textstyle k_{1}}
1455:{\displaystyle {m+n \choose r}.}
730: = 0 for all integers
715: = 0 for all integers
38:) is the following identity for
3242:{\displaystyle \Gamma (n+1)=n!}
2378:elements out of a first set of
1764:{\displaystyle {\binom {m}{k}}}
459:In general, the product of two
188:Alexandre-Théophile Vandermonde
186:. The identity is named after
3306:
3296:
3227:
3215:
3147:
3135:
3129:
3117:
3109:
3091:
3085:
3079:
3067:
3043:
3013:Gauss's hypergeometric theorem
2971:
2964:
2955:
2948:
2893:
2880:
2838:
2825:
2799:
2786:
2760:
2747:
2585:
2582:Chu–Vandermonde identity
2576:Chu–Vandermonde identity
1644:
1626:
1035:
1022:
1013:
1000:
975:
962:
765:
752:
1:
3469:Factorial and binomial topics
2728:and any non-negative integer
2502:sets. One therefore chooses
2349:On the one hand, one chooses
1361:By comparing coefficients of
2517:{\displaystyle \textstyle m}
2495:{\displaystyle \textstyle p}
2473:{\displaystyle \textstyle m}
2458:such sets, until a total of
2451:{\displaystyle \textstyle p}
471:, respectively, is given by
3385:hypergeometric distribution
3194:{\displaystyle \;_{2}F_{1}}
2856:{\displaystyle (1+x)^{s+t}}
2586:Askey 1975, pp. 59–60
1585:Take a rectangular grid of
207:to this theorem called the
3490:
3261:and applying the identity
18:
2811:{\displaystyle (1+x)^{t}}
2772:{\displaystyle (1+x)^{s}}
36:Vandermonde's convolution
3381:probability distribution
2105:
1799:). Similarly, there are
3395:from an urn containing
3203:hypergeometric function
3355:
3243:
3195:
3157:
2990:
2925:
2857:
2812:
2773:
2708:
2651:
2566:
2518:
2496:
2474:
2452:
2430:
2401:
2372:
2333:
2000:
1840:
1765:
1696:
1571:
1511:
1456:
1381:. For larger integers
1333:
1252:
1224:
1148:
1081:
914:
857:
809:
699:
648:
620:
563:
508:
440:
162:
105:
32:Vandermonde's identity
3418:Hockey-stick identity
3356:
3244:
3196:
3158:
2991:
2905:
2858:
2813:
2774:
2709:
2631:
2588:) and takes the form
2567:
2519:
2497:
2475:
2453:
2431:
2402:
2373:
2334:
2001:
1841:
1766:
1697:
1601:) squares. There are
1572:
1491:
1457:
1395:double counting proof
1334:
1232:
1198:
1128:
1061:
888:
858:
783:
700:
628:
594:
543:
488:
441:
212:-Vandermonde identity
192:Chinese mathematician
163:
85:
40:binomial coefficients
3474:Algebraic identities
3423:Rothe–Hagen identity
3369:Rothe–Hagen identity
3268:
3209:
3170:
3022:
3015:, which states that
2877:
2822:
2783:
2744:
2595:
2528:
2506:
2484:
2462:
2440:
2411:
2382:
2353:
2117:
1941:
1803:
1736:
1608:
1488:
1416:
881:
749:
478:
224:
171:for any nonnegative
49:
3393:without replacement
2025:), and go through (
1846:paths starting at (
1389:Combinatorial proof
1369:with 0 ≤
3351:
3239:
3191:
3153:
2986:
2868:Pochhammer symbols
2853:
2808:
2769:
2704:
2562:
2561:
2514:
2513:
2492:
2491:
2470:
2469:
2448:
2447:
2426:
2425:
2397:
2396:
2368:
2367:
2329:
2161:
1996:
1836:
1761:
1692:
1567:
1452:
1329:
1327:
853:
695:
436:
321:
158:
21:Vandermonde matrix
3413:Pascal's identity
3343:
3285:
3151:
2940:
2696:
2666:
2620:
2318:
2265:
2226:
2190:
2120:
1988:
1958:
1934:. Thus there are
1882:right moves and (
1874:), as a total of
1828:
1753:
1684:
1651:
1581:Geometrical proof
1556:
1526:
1441:
1297:
1267:
1163:
1096:
937:
832:
425:
386:
350:
280:
269:
150:
120:
74:
3481:
3453:
3452:for the history.
3451:
3439:
3360:
3358:
3357:
3352:
3350:
3349:
3348:
3339:
3321:
3314:
3313:
3292:
3291:
3290:
3277:
3248:
3246:
3245:
3240:
3200:
3198:
3197:
3192:
3190:
3189:
3180:
3179:
3162:
3160:
3159:
3154:
3152:
3150:
3112:
3074:
3042:
3041:
3032:
3031:
3005:binomial theorem
2995:
2993:
2992:
2987:
2985:
2984:
2963:
2962:
2947:
2946:
2945:
2932:
2924:
2919:
2901:
2900:
2862:
2860:
2859:
2854:
2852:
2851:
2817:
2815:
2814:
2809:
2807:
2806:
2778:
2776:
2775:
2770:
2768:
2767:
2713:
2711:
2710:
2705:
2703:
2702:
2701:
2695:
2680:
2673:
2672:
2671:
2658:
2650:
2645:
2627:
2626:
2625:
2616:
2604:
2571:
2569:
2568:
2563:
2560:
2559:
2541:
2540:
2524:elements out of
2523:
2521:
2520:
2515:
2501:
2499:
2498:
2493:
2479:
2477:
2476:
2471:
2457:
2455:
2454:
2449:
2435:
2433:
2432:
2427:
2424:
2423:
2406:
2404:
2403:
2398:
2395:
2394:
2377:
2375:
2374:
2369:
2366:
2365:
2338:
2336:
2335:
2330:
2325:
2324:
2323:
2314:
2313:
2312:
2294:
2293:
2282:
2272:
2271:
2270:
2264:
2263:
2254:
2253:
2243:
2233:
2232:
2231:
2225:
2224:
2215:
2214:
2204:
2197:
2196:
2195:
2189:
2188:
2179:
2178:
2168:
2160:
2153:
2152:
2134:
2133:
2005:
2003:
2002:
1997:
1995:
1994:
1993:
1987:
1972:
1965:
1964:
1963:
1950:
1845:
1843:
1842:
1837:
1835:
1834:
1833:
1827:
1812:
1787:right moves and
1770:
1768:
1767:
1762:
1760:
1759:
1758:
1745:
1709:right moves and
1701:
1699:
1698:
1693:
1691:
1690:
1689:
1680:
1668:
1658:
1657:
1656:
1647:
1617:
1576:
1574:
1573:
1568:
1563:
1562:
1561:
1555:
1540:
1533:
1532:
1531:
1518:
1510:
1505:
1461:
1459:
1458:
1453:
1448:
1447:
1446:
1437:
1425:
1358:, respectively.
1354: >
1346: >
1338:
1336:
1335:
1330:
1328:
1321:
1320:
1311:
1310:
1304:
1303:
1302:
1296:
1281:
1274:
1273:
1272:
1259:
1251:
1246:
1231:
1230:
1223:
1212:
1191:
1187:
1186:
1180:
1179:
1170:
1169:
1168:
1155:
1147:
1142:
1127:
1126:
1120:
1119:
1113:
1112:
1103:
1102:
1101:
1088:
1080:
1075:
1060:
1059:
1047:
1043:
1042:
1021:
1020:
993:
989:
988:
954:
953:
944:
943:
942:
933:
921:
913:
902:
862:
860:
859:
854:
849:
848:
839:
838:
837:
828:
816:
808:
797:
779:
778:
740:binomial theorem
734: >
719: >
704:
702:
701:
696:
691:
690:
681:
680:
674:
673:
658:
657:
647:
642:
627:
626:
619:
608:
590:
589:
583:
582:
573:
572:
562:
557:
542:
541:
535:
534:
528:
527:
518:
517:
507:
502:
487:
486:
445:
443:
442:
437:
432:
431:
430:
424:
423:
414:
413:
403:
393:
392:
391:
385:
384:
375:
374:
364:
357:
356:
355:
349:
348:
339:
338:
328:
320:
313:
312:
294:
293:
276:
275:
274:
265:
264:
263:
245:
244:
233:
167:
165:
164:
159:
157:
156:
155:
149:
134:
127:
126:
125:
112:
104:
99:
81:
80:
79:
70:
58:
3489:
3488:
3484:
3483:
3482:
3480:
3479:
3478:
3459:
3458:
3457:
3456:
3442:
3440:
3436:
3431:
3409:
3377:
3323:
3316:
3305:
3272:
3266:
3265:
3207:
3206:
3181:
3173:
3168:
3167:
3113:
3075:
3033:
3025:
3020:
3019:
3003:variant of the
2970:
2954:
2927:
2892:
2875:
2874:
2837:
2820:
2819:
2798:
2781:
2780:
2759:
2742:
2741:
2738:binomial series
2685:
2675:
2653:
2606:
2599:
2593:
2592:
2578:
2551:
2532:
2526:
2525:
2504:
2503:
2482:
2481:
2460:
2459:
2438:
2437:
2415:
2409:
2408:
2407:elements; then
2386:
2380:
2379:
2357:
2351:
2350:
2344:double counting
2304:
2285:
2284:
2277:
2255:
2245:
2238:
2216:
2206:
2199:
2180:
2170:
2163:
2144:
2125:
2115:
2114:
2108:
2103:
2101:Generalizations
2069:(as the point (
2057:), so sum from
1977:
1967:
1945:
1939:
1938:
1858:) that end at (
1817:
1807:
1801:
1800:
1740:
1734:
1733:
1725: +
1670:
1663:
1619:
1612:
1606:
1605:
1583:
1545:
1535:
1513:
1486:
1485:
1477: −
1427:
1420:
1414:
1413:
1391:
1377: +
1326:
1325:
1312:
1286:
1276:
1254:
1189:
1188:
1171:
1150:
1104:
1083:
1045:
1044:
1034:
1012:
991:
990:
974:
955:
945:
923:
916:
879:
878:
840:
818:
811:
764:
747:
746:
728:
713:
682:
659:
649:
574:
564:
519:
509:
476:
475:
457:
455:Algebraic proof
452:
415:
405:
398:
376:
366:
359:
340:
330:
323:
304:
285:
255:
236:
235:
228:
222:
221:
139:
129:
107:
60:
53:
47:
46:
24:
17:
12:
11:
5:
3487:
3485:
3477:
3476:
3471:
3461:
3460:
3455:
3454:
3444:Askey, Richard
3433:
3432:
3430:
3427:
3426:
3425:
3420:
3415:
3408:
3405:
3403:blue marbles.
3376:
3373:
3362:
3361:
3347:
3342:
3338:
3335:
3332:
3329:
3326:
3320:
3312:
3308:
3304:
3301:
3298:
3295:
3289:
3284:
3281:
3276:
3257: = −
3251:gamma function
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3188:
3184:
3178:
3164:
3163:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3040:
3036:
3030:
2997:
2996:
2983:
2980:
2977:
2973:
2969:
2966:
2961:
2957:
2953:
2950:
2944:
2939:
2936:
2931:
2923:
2918:
2915:
2912:
2908:
2904:
2899:
2895:
2891:
2888:
2885:
2882:
2850:
2847:
2844:
2840:
2836:
2833:
2830:
2827:
2805:
2801:
2797:
2794:
2791:
2788:
2766:
2762:
2758:
2755:
2752:
2749:
2719:complex-valued
2715:
2714:
2700:
2694:
2691:
2688:
2684:
2679:
2670:
2665:
2662:
2657:
2649:
2644:
2641:
2638:
2634:
2630:
2624:
2619:
2615:
2612:
2609:
2603:
2577:
2574:
2558:
2554:
2550:
2547:
2544:
2539:
2535:
2512:
2490:
2468:
2446:
2422:
2418:
2393:
2389:
2364:
2360:
2340:
2339:
2328:
2322:
2317:
2311:
2307:
2303:
2300:
2297:
2292:
2288:
2281:
2275:
2269:
2262:
2258:
2252:
2248:
2242:
2236:
2230:
2223:
2219:
2213:
2209:
2203:
2194:
2187:
2183:
2177:
2173:
2167:
2159:
2156:
2151:
2147:
2143:
2140:
2137:
2132:
2128:
2123:
2107:
2104:
2102:
2099:
2007:
2006:
1992:
1986:
1983:
1980:
1976:
1971:
1962:
1957:
1954:
1949:
1832:
1826:
1823:
1820:
1816:
1811:
1757:
1752:
1749:
1744:
1703:
1702:
1688:
1683:
1679:
1676:
1673:
1667:
1661:
1655:
1650:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1616:
1582:
1579:
1578:
1577:
1566:
1560:
1554:
1551:
1548:
1544:
1539:
1530:
1525:
1522:
1517:
1509:
1504:
1501:
1498:
1494:
1463:
1462:
1451:
1445:
1440:
1436:
1433:
1430:
1424:
1390:
1387:
1340:
1339:
1324:
1319:
1315:
1309:
1301:
1295:
1292:
1289:
1285:
1280:
1271:
1266:
1263:
1258:
1250:
1245:
1242:
1239:
1235:
1229:
1222:
1219:
1216:
1211:
1208:
1205:
1201:
1197:
1194:
1192:
1190:
1185:
1178:
1174:
1167:
1162:
1159:
1154:
1146:
1141:
1138:
1135:
1131:
1125:
1118:
1111:
1107:
1100:
1095:
1092:
1087:
1079:
1074:
1071:
1068:
1064:
1058:
1053:
1050:
1048:
1046:
1041:
1037:
1033:
1030:
1027:
1024:
1019:
1015:
1011:
1008:
1005:
1002:
999:
996:
994:
992:
987:
984:
981:
977:
973:
970:
967:
964:
961:
958:
956:
952:
948:
941:
936:
932:
929:
926:
920:
912:
909:
906:
901:
898:
895:
891:
887:
886:
864:
863:
852:
847:
843:
836:
831:
827:
824:
821:
815:
807:
804:
801:
796:
793:
790:
786:
782:
777:
774:
771:
767:
763:
760:
757:
754:
726:
711:
706:
705:
694:
689:
685:
679:
672:
669:
666:
662:
656:
652:
646:
641:
638:
635:
631:
625:
618:
615:
612:
607:
604:
601:
597:
593:
588:
581:
577:
571:
567:
561:
556:
553:
550:
546:
540:
533:
526:
522:
516:
512:
506:
501:
498:
495:
491:
485:
456:
453:
451:
448:
447:
446:
435:
429:
422:
418:
412:
408:
402:
396:
390:
383:
379:
373:
369:
363:
354:
347:
343:
337:
333:
327:
319:
316:
311:
307:
303:
300:
297:
292:
288:
283:
279:
273:
268:
262:
258:
254:
251:
248:
243:
239:
232:
169:
168:
154:
148:
145:
142:
138:
133:
124:
119:
116:
111:
103:
98:
95:
92:
88:
84:
78:
73:
69:
66:
63:
57:
15:
13:
10:
9:
6:
4:
3:
2:
3486:
3475:
3472:
3470:
3467:
3466:
3464:
3449:
3445:
3438:
3435:
3428:
3424:
3421:
3419:
3416:
3414:
3411:
3410:
3406:
3404:
3402:
3398:
3394:
3390:
3386:
3382:
3374:
3372:
3370:
3365:
3340:
3336:
3333:
3330:
3327:
3324:
3310:
3302:
3299:
3293:
3282:
3279:
3264:
3263:
3262:
3260:
3256:
3252:
3236:
3233:
3230:
3224:
3221:
3218:
3204:
3186:
3182:
3176:
3144:
3141:
3138:
3126:
3123:
3120:
3106:
3103:
3100:
3097:
3094:
3082:
3070:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3038:
3034:
3028:
3018:
3017:
3016:
3014:
3010:
3009:binomial type
3006:
3002:
2981:
2978:
2975:
2967:
2959:
2951:
2937:
2934:
2921:
2916:
2913:
2910:
2906:
2902:
2897:
2889:
2886:
2883:
2873:
2872:
2871:
2869:
2864:
2848:
2845:
2842:
2834:
2831:
2828:
2803:
2795:
2792:
2789:
2764:
2756:
2753:
2750:
2739:
2735:
2731:
2727:
2723:
2720:
2692:
2689:
2686:
2682:
2663:
2660:
2647:
2642:
2639:
2636:
2632:
2628:
2617:
2613:
2610:
2607:
2591:
2590:
2589:
2587:
2583:
2575:
2573:
2556:
2552:
2548:
2545:
2542:
2537:
2533:
2510:
2488:
2466:
2444:
2420:
2416:
2391:
2387:
2362:
2358:
2347:
2345:
2326:
2315:
2309:
2305:
2301:
2298:
2295:
2290:
2286:
2273:
2260:
2256:
2250:
2246:
2234:
2221:
2217:
2211:
2207:
2185:
2181:
2175:
2171:
2157:
2154:
2149:
2145:
2141:
2138:
2135:
2130:
2126:
2121:
2113:
2112:
2111:
2100:
2098:
2096:
2092:
2088:
2084:
2080:
2076:
2072:
2068:
2064:
2060:
2056:
2052:
2048:
2044:
2040:
2037:). This is a
2036:
2032:
2028:
2024:
2020:
2016:
2012:
1984:
1981:
1978:
1974:
1955:
1952:
1937:
1936:
1935:
1933:
1929:
1925:
1921:
1917:
1913:
1909:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1873:
1869:
1865:
1861:
1857:
1853:
1849:
1824:
1821:
1818:
1814:
1798:
1794:
1790:
1786:
1782:
1778:
1774:
1750:
1747:
1730:
1728:
1724:
1720:
1716:
1712:
1708:
1681:
1677:
1674:
1671:
1659:
1648:
1641:
1638:
1635:
1632:
1629:
1623:
1620:
1604:
1603:
1602:
1600:
1596:
1592:
1588:
1580:
1564:
1552:
1549:
1546:
1542:
1523:
1520:
1507:
1502:
1499:
1496:
1492:
1484:
1483:
1482:
1480:
1476:
1472:
1468:
1449:
1438:
1434:
1431:
1428:
1412:
1411:
1410:
1408:
1404:
1400:
1396:
1388:
1386:
1384:
1380:
1376:
1373: ≤
1372:
1368:
1364:
1359:
1357:
1353:
1349:
1345:
1322:
1317:
1313:
1293:
1290:
1287:
1283:
1264:
1261:
1248:
1243:
1240:
1237:
1233:
1220:
1217:
1214:
1209:
1206:
1203:
1199:
1195:
1193:
1176:
1172:
1160:
1157:
1144:
1139:
1136:
1133:
1129:
1109:
1105:
1093:
1090:
1077:
1072:
1069:
1066:
1062:
1051:
1049:
1039:
1031:
1028:
1025:
1017:
1009:
1006:
1003:
997:
995:
985:
982:
979:
971:
968:
965:
959:
957:
950:
946:
934:
930:
927:
924:
910:
907:
904:
899:
896:
893:
889:
877:
876:
875:
873:
869:
850:
845:
841:
829:
825:
822:
819:
805:
802:
799:
794:
791:
788:
784:
780:
775:
772:
769:
761:
758:
755:
745:
744:
743:
741:
737:
733:
729:
722:
718:
714:
692:
687:
683:
670:
667:
664:
660:
654:
650:
644:
639:
636:
633:
629:
616:
613:
610:
605:
602:
599:
595:
591:
579:
575:
569:
565:
559:
554:
551:
548:
544:
524:
520:
514:
510:
504:
499:
496:
493:
489:
474:
473:
472:
470:
466:
463:with degrees
462:
454:
449:
433:
420:
416:
410:
406:
394:
381:
377:
371:
367:
345:
341:
335:
331:
317:
314:
309:
305:
301:
298:
295:
290:
286:
281:
277:
266:
260:
256:
252:
249:
246:
241:
237:
220:
219:
218:
215:
213:
211:
206:
204:
198:
196:
193:
189:
185:
181:
177:
174:
146:
143:
140:
136:
117:
114:
101:
96:
93:
90:
86:
82:
71:
67:
64:
61:
45:
44:
43:
41:
37:
33:
29:
28:combinatorics
22:
3447:
3437:
3400:
3396:
3392:
3388:
3378:
3366:
3363:
3258:
3254:
3165:
2998:
2865:
2729:
2725:
2721:
2717:for general
2716:
2581:
2579:
2348:
2341:
2109:
2094:
2090:
2086:
2082:
2078:
2074:
2070:
2066:
2062:
2058:
2054:
2050:
2046:
2042:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2008:
1931:
1927:
1923:
1919:
1915:
1911:
1907:
1903:
1899:
1895:
1891:
1887:
1883:
1879:
1875:
1871:
1867:
1863:
1859:
1855:
1851:
1847:
1796:
1792:
1788:
1784:
1780:
1776:
1772:
1731:
1726:
1722:
1718:
1714:
1710:
1706:
1704:
1598:
1594:
1590:
1586:
1584:
1478:
1474:
1470:
1466:
1464:
1406:
1402:
1398:
1392:
1382:
1378:
1374:
1370:
1366:
1362:
1360:
1355:
1351:
1347:
1343:
1341:
871:
867:
865:
735:
731:
724:
720:
716:
709:
707:
468:
464:
458:
216:
209:
202:
199:
183:
179:
175:
170:
35:
31:
25:
3364:liberally.
2734:multiplying
461:polynomials
200:There is a
3463:Categories
3429:References
2346:argument.
1732:There are
195:Zhu Shijie
3334:−
3328:−
3300:−
3213:Γ
3142:−
3133:Γ
3124:−
3115:Γ
3104:−
3098:−
3089:Γ
3077:Γ
2979:−
2907:∑
2690:−
2633:∑
2546:⋯
2299:⋯
2235:⋯
2139:⋯
2122:∑
1982:−
1822:−
1639:−
1550:−
1493:∑
1291:−
1234:∑
1200:∑
1130:∑
1063:∑
890:∑
785:∑
738:. By the
668:−
630:∑
596:∑
545:∑
490:∑
395:⋯
299:⋯
282:∑
250:⋯
144:−
87:∑
3446:(1975),
3407:See also
3399:red and
1473:men and
1401:men and
173:integers
3383:is the
3249:is the
3201:is the
2061:= 0 to
1481:women:
205:-analog
3391:draws
3166:where
3001:umbral
2039:subset
1783:), as
450:Proofs
2584:(see
1922:) − (
1894:) − (
3441:See
3367:The
3205:and
2779:and
2740:for
2736:the
2724:and
1930:) =
1350:and
870:and
723:and
467:and
34:(or
2870:as
2097:).
1910:+ (
1589:x (
26:In
3465::
2863:.
2085:,
2073:,
2065:=
2045:,
2029:,
2013:,
1862:,
1850:,
1775:,
742:,
214:.
197:.
182:,
178:,
42::
30:,
3401:m
3397:n
3389:r
3346:)
3341:k
3337:1
3331:n
3325:k
3319:(
3311:k
3307:)
3303:1
3297:(
3294:=
3288:)
3283:k
3280:n
3275:(
3259:n
3255:a
3237:!
3234:n
3231:=
3228:)
3225:1
3222:+
3219:n
3216:(
3187:1
3183:F
3177:2
3148:)
3145:b
3139:c
3136:(
3130:)
3127:a
3121:c
3118:(
3110:)
3107:b
3101:a
3095:c
3092:(
3086:)
3083:c
3080:(
3071:=
3068:)
3065:1
3062:;
3059:c
3056:;
3053:b
3050:,
3047:a
3044:(
3039:1
3035:F
3029:2
2982:k
2976:n
2972:)
2968:t
2965:(
2960:k
2956:)
2952:s
2949:(
2943:)
2938:k
2935:n
2930:(
2922:n
2917:0
2914:=
2911:k
2903:=
2898:n
2894:)
2890:t
2887:+
2884:s
2881:(
2849:t
2846:+
2843:s
2839:)
2835:x
2832:+
2829:1
2826:(
2804:t
2800:)
2796:x
2793:+
2790:1
2787:(
2765:s
2761:)
2757:x
2754:+
2751:1
2748:(
2730:n
2726:t
2722:s
2699:)
2693:k
2687:n
2683:t
2678:(
2669:)
2664:k
2661:s
2656:(
2648:n
2643:0
2640:=
2637:k
2629:=
2623:)
2618:n
2614:t
2611:+
2608:s
2602:(
2557:p
2553:n
2549:+
2543:+
2538:1
2534:n
2511:m
2489:p
2467:m
2445:p
2421:2
2417:k
2392:1
2388:n
2363:1
2359:k
2327:.
2321:)
2316:m
2310:p
2306:n
2302:+
2296:+
2291:1
2287:n
2280:(
2274:=
2268:)
2261:p
2257:k
2251:p
2247:n
2241:(
2229:)
2222:2
2218:k
2212:2
2208:n
2202:(
2193:)
2186:1
2182:k
2176:1
2172:n
2166:(
2158:m
2155:=
2150:p
2146:k
2142:+
2136:+
2131:1
2127:k
2095:r
2093:−
2091:n
2089:+
2087:m
2083:r
2079:k
2077:−
2075:m
2071:k
2067:r
2063:k
2059:k
2055:r
2053:−
2051:n
2049:+
2047:m
2043:r
2035:k
2033:−
2031:m
2027:k
2023:r
2021:−
2019:n
2017:+
2015:m
2011:r
1991:)
1985:k
1979:r
1975:n
1970:(
1961:)
1956:k
1953:m
1948:(
1932:n
1928:k
1926:−
1924:m
1920:r
1918:−
1916:n
1914:+
1912:m
1908:k
1906:−
1904:r
1900:k
1898:−
1896:m
1892:r
1890:−
1888:n
1886:+
1884:m
1880:k
1878:−
1876:r
1872:r
1870:−
1868:n
1866:+
1864:m
1860:r
1856:k
1854:−
1852:m
1848:k
1831:)
1825:k
1819:r
1815:n
1810:(
1797:m
1793:k
1791:−
1789:m
1785:k
1781:k
1779:−
1777:m
1773:k
1756:)
1751:k
1748:m
1743:(
1727:n
1723:m
1719:r
1717:-
1715:n
1713:+
1711:m
1707:r
1687:)
1682:r
1678:n
1675:+
1672:m
1666:(
1660:=
1654:)
1649:r
1645:)
1642:r
1636:n
1633:+
1630:m
1627:(
1624:+
1621:r
1615:(
1599:r
1597:−
1595:n
1593:+
1591:m
1587:r
1565:.
1559:)
1553:k
1547:r
1543:n
1538:(
1529:)
1524:k
1521:m
1516:(
1508:r
1503:0
1500:=
1497:k
1479:k
1475:r
1471:k
1467:k
1450:.
1444:)
1439:r
1435:n
1432:+
1429:m
1423:(
1407:r
1403:n
1399:m
1383:r
1379:n
1375:m
1371:r
1367:r
1363:x
1356:n
1352:j
1348:m
1344:i
1323:,
1318:r
1314:x
1308:)
1300:)
1294:k
1288:r
1284:n
1279:(
1270:)
1265:k
1262:m
1257:(
1249:r
1244:0
1241:=
1238:k
1228:(
1221:n
1218:+
1215:m
1210:0
1207:=
1204:r
1196:=
1184:)
1177:j
1173:x
1166:)
1161:j
1158:n
1153:(
1145:n
1140:0
1137:=
1134:j
1124:(
1117:)
1110:i
1106:x
1099:)
1094:i
1091:m
1086:(
1078:m
1073:0
1070:=
1067:i
1057:(
1052:=
1040:n
1036:)
1032:x
1029:+
1026:1
1023:(
1018:m
1014:)
1010:x
1007:+
1004:1
1001:(
998:=
986:n
983:+
980:m
976:)
972:x
969:+
966:1
963:(
960:=
951:r
947:x
940:)
935:r
931:n
928:+
925:m
919:(
911:n
908:+
905:m
900:0
897:=
894:r
872:n
868:m
851:.
846:r
842:x
835:)
830:r
826:n
823:+
820:m
814:(
806:n
803:+
800:m
795:0
792:=
789:r
781:=
776:n
773:+
770:m
766:)
762:x
759:+
756:1
753:(
736:n
732:j
727:j
725:b
721:m
717:i
712:i
710:a
693:,
688:r
684:x
678:)
671:k
665:r
661:b
655:k
651:a
645:r
640:0
637:=
634:k
624:(
617:n
614:+
611:m
606:0
603:=
600:r
592:=
587:)
580:j
576:x
570:j
566:b
560:n
555:0
552:=
549:j
539:(
532:)
525:i
521:x
515:i
511:a
505:m
500:0
497:=
494:i
484:(
469:n
465:m
434:.
428:)
421:p
417:k
411:p
407:n
401:(
389:)
382:2
378:k
372:2
368:n
362:(
353:)
346:1
342:k
336:1
332:n
326:(
318:m
315:=
310:p
306:k
302:+
296:+
291:1
287:k
278:=
272:)
267:m
261:p
257:n
253:+
247:+
242:1
238:n
231:(
210:q
203:q
184:n
180:m
176:r
153:)
147:k
141:r
137:n
132:(
123:)
118:k
115:m
110:(
102:r
97:0
94:=
91:k
83:=
77:)
72:r
68:n
65:+
62:m
56:(
23:.
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