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Update: I've cleaned up the notation somewhat; it would be nice if someone can proofread this and make sure I haven't introduced any silly errors. There seems to be some confusion as to whether the "Cauchy product" refers to the sequence or the series or both. It bothers me that the lead paragraph
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I agree that the formula is correct: the left hand side equals the right hand side. There is no problem there. The issue is that I don't think it belongs on this page: I don't believe that it's actually called "Cauchy product". The point of the Cauchy product for infinite series is that when you
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Cesaro's theorem is currently unsourced. It traces back to Cesaro. See
Silverman's book "On the Definition of the Sum of a Divergent Series", p.27, Thm. J for a modern English statement. He cites "Cesaro: Bull des Sciences math, t. XIV 1890". I do not presently have an inclination to figure out
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tends to infinity, the right-hand side converges to the Cauchy product. As formal power series, convergence in the adic topology is trivial. Convergence in the usual topology (as a series of real numbers, not as a formal power series) is not always true. Indeed, the article already gives the
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I've removed the "unreferenced" tag and added a reference to
Apostol. This includes most of the content, but not Cesaro's theorem; a further reference would be good. I've tagged the article "attention needed", as I find the notation C(x,y) somewhat confusing, and the use of
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I don't think this formula is relevant to this article. It should probably be in some other article on finite series. My objection is that this formula isn't a Cauchy product; as far as I'm concerned, Cauchy products are only defined for infinite
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that I had deleted. In my view, this concept is too advanced for the main audience of this article: first-year B.S. students in mathematics or second-year B.S. students in engineering. The relation to semigroup algebras would make sense in article
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09:47, 22 November 2011 46.17.97.92 (talk) (9,053 bytes) (Undid revision 461850889 by Jowa fan (talk) This formula comes from the lessons of calculus from my university (moscow state university) in elec.engin. Why it is a nonsense?) (undo)
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and the section "series" seem to say much the same; can anyone think of a nice way of rewriting the lead without using mathematical notation? The sections "Generalizations" and "Relation to convolution of functions" seem to be unsourced.
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I changed the formulation of Cesaro's theorem. I believe the theorem was intended to be read in a way that would make it corect but there was a notational conflict with the rest of the article rendering it liable to be misunderstood.
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It is quite interesting the fact that if n goes to infinity this summation product become the series Cauchy product. ________________________________________________________________________________________________
841:{\displaystyle \left(\sum _{k=0}^{n}a_{k}\right)\cdot \left(\sum _{k=0}^{n}b_{k}\right)=\sum _{k=0}^{2n}\sum _{i=0}^{k}a_{i}b_{k-i}-\sum _{k=0}^{n-1}(a_{k}\sum _{i=n+1}^{2n-k}b_{i}+b_{k}\sum _{i=n+1}^{2n-k}a_{i})}
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13:24, 22 November 2011 Jowa fan (talk | contribs) (8,534 bytes) (Undid revision 461914058 by 46.17.97.92 (talk) 1. It's not clear what c_k equals; 2. See policy at WP:SOURCE) (undo)
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23:41, 21 November 2011 Jowa fan (talk | contribs) (8,534 bytes) (Undid revision 461802692 by 46.17.97.92 (talk) remove new section: no sources, and it doesn't make sense) (undo)
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Hello Jowa fan, I think i will proofread the article today evening; my talk page indicates that it would be a good idea to finally do something useful here anyways. Good day, --
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It is interesting that if we increase n by 1 all the formula "shifts" by one row and it is not necessary to delete the two "side" terms on k=4. We need to delete from k=5.
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Who decided the scope of the article? I understand the engineering students may not be interested in semigroup algebras; those readers should just skip the section. --
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strikes me as too informal. I'm not sure whether it would be better to give a clear definition of "C(x,y)", or rewrite the article to avoid using that notation.
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I also changed
Spanish entry and noticed that the French and German entries have essentially the same counterexample, with brief proofs of the divergence.
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for your formula. I agree that the formula is correct; the part that doesn't make sense to me is whether it belongs on this page.
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Knowledge's citation system enough to edit the article with this information, but I wanted to at least put it somewhere public.
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I'm editing the article to remove these statements. Can anyone think of a better home for the remaining formula?
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and whose Cauchy square does converge, though conditionally, contrary to the stated. I put the counterexample
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Because of the minus in front of the second piece, this delete the superabundant terms of the first piece.
2076:{\displaystyle k=6\Rightarrow a_{0}b_{6}+a_{1}b_{5}+a_{2}b_{4}+a_{3}b_{3}+a_{4}b_{2}+a_{5}b_{1}+a_{6}b_{0}}
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In addition, I believe that some of the text is wrong. The article claims that it's interesting that as
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sum an infinite series, the answer may depend on the order of summation. So the Cauchy product defines
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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________________________________________________________________________________________________
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2393:{\displaystyle k=0\Rightarrow a_{0}b_{4}+a_{0}b_{5}+a_{0}b_{6}+b_{0}a_{4}+b_{0}a_{5}+b_{0}a_{6}}
1888:{\displaystyle k=5\Rightarrow a_{0}b_{5}+a_{1}b_{4}+a_{2}b_{3}+a_{3}b_{2}+a_{4}b_{1}+a_{5}b_{0}}
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If the product comes from two finite summations, up to n, the Cauchy
Product is defined as
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1723:{\displaystyle k=4\Rightarrow a_{0}b_{4}+a_{1}b_{3}+a_{2}b_{2}+a_{3}b_{1}+a_{4}b_{0}}
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I took a piece of paper and I tried to puzzle over the formula. I made the case n=3.
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1581:{\displaystyle k=3\Rightarrow a_{0}b_{3}+a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{0}}
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The beautifulness of a mathematical formula is that anyone can verify it.
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In these days I was bustling about with this formulas so I arrived here.
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If n→∞ we don't need to cancel anything and we have the Cauchy product.
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321:{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{\sqrt {n+1}}}}
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In my opinion this could be "wikipediafied". What do you think?
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If n is increased by 2 we'll start to delete from k=6 and so on.
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fixing but issues. It is well defined and its Cauchy square
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The formula comes from lesson notes I took at university.
221:{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n}}}
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is that I want to verify whether anyone uses the name
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2237:
2207:
2170:
2157:
2120:
2091:
2090:
2063:
2053:
2040:
2030:
2017:
2007:
1994:
1984:
1971:
1961:
1948:
1938:
1925:
1915:
1898:
1897:
1875:
1865:
1852:
1842:
1829:
1819:
1806:
1796:
1783:
1773:
1760:
1750:
1733:
1732:
1710:
1700:
1687:
1677:
1664:
1654:
1641:
1631:
1618:
1608:
1591:
1590:
1568:
1558:
1545:
1535:
1522:
1512:
1499:
1489:
1472:
1471:
1449:
1439:
1426:
1416:
1403:
1393:
1376:
1375:
1353:
1343:
1330:
1320:
1303:
1302:
1280:
1270:
1253:
1252:
1219:
1209:
1162:
1161:
1131:
1118:
1105:
1092:
1076:
1063:
1050:
1037:
1029:
1028:
996:
974:
970:
952:
930:
926:
921:
920:
863:
856:
825:
779:
766:
720:
671:
661:
598:
576:
572:
554:
532:
528:
523:
522:
510:
441:
440:
436:
416:
365:
360:
359:
338:
330:
329:
294:
284:
256:
255:
230:
229:
202:
192:
164:
163:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
3098:
3096:
3088:
3087:
3082:
3072:
3071:
3052:
3049:
3048:
3047:
3001:
2998:
2997:
2996:
2995:
2994:
2993:
2992:
2969:
2968:
2967:
2966:
2949:
2948:
2947:
2946:
2932:
2929:
2926:
2920:
2914:
2911:
2908:
2904:
2900:
2897:
2894:
2891:
2886:
2882:
2878:
2873:
2869:
2849:
2848:
2847:
2846:
2839:
2838:
2820:
2817:Cauchy product
2809:
2795:
2792:
2789:
2766:
2762:
2739:
2735:
2712:
2708:
2687:
2684:
2679:
2675:
2671:
2666:
2662:
2639:
2635:
2593:
2592:
2579:
2575:
2569:
2565:
2561:
2556:
2552:
2546:
2542:
2538:
2535:
2532:
2529:
2519:
2506:
2502:
2496:
2492:
2488:
2483:
2479:
2473:
2469:
2465:
2460:
2456:
2450:
2446:
2442:
2437:
2433:
2427:
2423:
2419:
2416:
2413:
2410:
2400:
2387:
2383:
2377:
2373:
2369:
2364:
2360:
2354:
2350:
2346:
2341:
2337:
2331:
2327:
2323:
2318:
2314:
2308:
2304:
2300:
2295:
2291:
2285:
2281:
2277:
2272:
2268:
2262:
2258:
2254:
2251:
2248:
2245:
2231:
2230:
2219:
2214:
2210:
2204:
2201:
2198:
2193:
2190:
2187:
2183:
2177:
2173:
2169:
2164:
2160:
2154:
2151:
2148:
2143:
2140:
2137:
2133:
2127:
2123:
2119:
2114:
2109:
2106:
2103:
2099:
2084:
2083:
2070:
2066:
2060:
2056:
2052:
2047:
2043:
2037:
2033:
2029:
2024:
2020:
2014:
2010:
2006:
2001:
1997:
1991:
1987:
1983:
1978:
1974:
1968:
1964:
1960:
1955:
1951:
1945:
1941:
1937:
1932:
1928:
1922:
1918:
1914:
1911:
1908:
1905:
1895:
1882:
1878:
1872:
1868:
1864:
1859:
1855:
1849:
1845:
1841:
1836:
1832:
1826:
1822:
1818:
1813:
1809:
1803:
1799:
1795:
1790:
1786:
1780:
1776:
1772:
1767:
1763:
1757:
1753:
1749:
1746:
1743:
1740:
1730:
1717:
1713:
1707:
1703:
1699:
1694:
1690:
1684:
1680:
1676:
1671:
1667:
1661:
1657:
1653:
1648:
1644:
1638:
1634:
1630:
1625:
1621:
1615:
1611:
1607:
1604:
1601:
1598:
1588:
1575:
1571:
1565:
1561:
1557:
1552:
1548:
1542:
1538:
1534:
1529:
1525:
1519:
1515:
1511:
1506:
1502:
1496:
1492:
1488:
1485:
1482:
1479:
1469:
1456:
1452:
1446:
1442:
1438:
1433:
1429:
1423:
1419:
1415:
1410:
1406:
1400:
1396:
1392:
1389:
1386:
1383:
1373:
1360:
1356:
1350:
1346:
1342:
1337:
1333:
1327:
1323:
1319:
1316:
1313:
1310:
1300:
1287:
1283:
1277:
1273:
1269:
1266:
1263:
1260:
1246:
1245:
1232:
1229:
1226:
1222:
1216:
1212:
1206:
1201:
1198:
1195:
1191:
1185:
1180:
1177:
1174:
1170:
1155:
1154:
1143:
1138:
1134:
1130:
1125:
1121:
1117:
1112:
1108:
1104:
1099:
1095:
1091:
1088:
1083:
1079:
1075:
1070:
1066:
1062:
1057:
1053:
1049:
1044:
1040:
1036:
1022:
1021:
1009:
1003:
999:
993:
988:
985:
982:
978:
973:
969:
965:
959:
955:
949:
944:
941:
938:
934:
929:
861:
854:
849:
848:
837:
832:
828:
822:
819:
816:
813:
808:
805:
802:
799:
796:
792:
786:
782:
778:
773:
769:
763:
760:
757:
754:
749:
746:
743:
740:
737:
733:
727:
723:
719:
714:
711:
708:
703:
700:
697:
693:
689:
684:
681:
678:
674:
668:
664:
658:
653:
650:
647:
643:
637:
634:
629:
626:
623:
619:
615:
611:
605:
601:
595:
590:
587:
584:
580:
575:
571:
567:
561:
557:
551:
546:
543:
540:
536:
531:
509:
506:
505:
504:
451:
448:
435:
432:
415:
412:
372:
368:
345:
341:
337:
314:
311:
308:
301:
297:
293:
290:
287:
279:
274:
271:
268:
264:
243:
240:
237:
215:
209:
205:
201:
198:
195:
187:
182:
179:
176:
172:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
3097:
3086:
3083:
3081:
3078:
3077:
3075:
3068:
3067:
3063:
3059:
3046:
3042:
3038:
3034:
3033:
3032:
3031:
3027:
3023:
3019:
3015:
3010:
3006:
2999:
2991:
2987:
2983:
2979:
2975:
2974:
2973:
2972:
2971:
2970:
2965:
2961:
2957:
2953:
2952:
2951:
2950:
2930:
2927:
2924:
2918:
2912:
2909:
2906:
2898:
2895:
2889:
2884:
2880:
2876:
2871:
2867:
2857:
2853:
2852:
2851:
2850:
2843:
2842:
2841:
2840:
2837:
2833:
2829:
2825:
2821:
2818:
2814:
2810:
2793:
2790:
2787:
2764:
2760:
2737:
2733:
2710:
2706:
2685:
2682:
2677:
2673:
2669:
2664:
2660:
2637:
2633:
2623:
2622:
2621:
2620:
2616:
2612:
2608:
2605:
2602:
2599:
2596:
2577:
2573:
2567:
2563:
2559:
2554:
2550:
2544:
2540:
2533:
2530:
2527:
2520:
2504:
2500:
2494:
2490:
2486:
2481:
2477:
2471:
2467:
2463:
2458:
2454:
2448:
2444:
2440:
2435:
2431:
2425:
2421:
2414:
2411:
2408:
2401:
2385:
2381:
2375:
2371:
2367:
2362:
2358:
2352:
2348:
2344:
2339:
2335:
2329:
2325:
2321:
2316:
2312:
2306:
2302:
2298:
2293:
2289:
2283:
2279:
2275:
2270:
2266:
2260:
2256:
2249:
2246:
2243:
2236:
2235:
2234:
2212:
2208:
2202:
2199:
2196:
2191:
2188:
2185:
2181:
2175:
2171:
2167:
2162:
2158:
2152:
2149:
2146:
2141:
2138:
2135:
2131:
2125:
2121:
2112:
2107:
2104:
2101:
2097:
2089:
2088:
2087:
2068:
2064:
2058:
2054:
2050:
2045:
2041:
2035:
2031:
2027:
2022:
2018:
2012:
2008:
2004:
1999:
1995:
1989:
1985:
1981:
1976:
1972:
1966:
1962:
1958:
1953:
1949:
1943:
1939:
1935:
1930:
1926:
1920:
1916:
1909:
1906:
1903:
1896:
1880:
1876:
1870:
1866:
1862:
1857:
1853:
1847:
1843:
1839:
1834:
1830:
1824:
1820:
1816:
1811:
1807:
1801:
1797:
1793:
1788:
1784:
1778:
1774:
1770:
1765:
1761:
1755:
1751:
1744:
1741:
1738:
1731:
1715:
1711:
1705:
1701:
1697:
1692:
1688:
1682:
1678:
1674:
1669:
1665:
1659:
1655:
1651:
1646:
1642:
1636:
1632:
1628:
1623:
1619:
1613:
1609:
1602:
1599:
1596:
1589:
1573:
1569:
1563:
1559:
1555:
1550:
1546:
1540:
1536:
1532:
1527:
1523:
1517:
1513:
1509:
1504:
1500:
1494:
1490:
1483:
1480:
1477:
1470:
1454:
1450:
1444:
1440:
1436:
1431:
1427:
1421:
1417:
1413:
1408:
1404:
1398:
1394:
1387:
1384:
1381:
1374:
1358:
1354:
1348:
1344:
1340:
1335:
1331:
1325:
1321:
1314:
1311:
1308:
1301:
1285:
1281:
1275:
1271:
1264:
1261:
1258:
1251:
1250:
1249:
1230:
1227:
1224:
1220:
1214:
1210:
1204:
1199:
1196:
1193:
1189:
1183:
1178:
1175:
1172:
1168:
1160:
1159:
1158:
1136:
1132:
1128:
1123:
1119:
1115:
1110:
1106:
1102:
1097:
1093:
1081:
1077:
1073:
1068:
1064:
1060:
1055:
1051:
1047:
1042:
1038:
1027:
1026:
1025:
1007:
1001:
997:
991:
986:
983:
980:
976:
971:
967:
963:
957:
953:
947:
942:
939:
936:
932:
927:
919:
918:
917:
914:
911:
908:
905:
902:
899:
898:
896:
892:
889:
888:
886:
881:
880:so I replied
878:
877:
873:
870:
866:
864:
857:
830:
826:
820:
817:
814:
811:
806:
803:
800:
797:
794:
790:
784:
780:
776:
771:
767:
761:
758:
755:
752:
747:
744:
741:
738:
735:
731:
725:
721:
712:
709:
706:
701:
698:
695:
691:
687:
682:
679:
676:
672:
666:
662:
656:
651:
648:
645:
641:
635:
632:
627:
624:
621:
617:
613:
609:
603:
599:
593:
588:
585:
582:
578:
573:
569:
565:
559:
555:
549:
544:
541:
538:
534:
529:
521:
520:
519:
516:
513:
507:
503:
499:
495:
491:
490:
489:
488:
484:
480:
474:
473:
469:
465:
449:
446:
433:
431:
430:
426:
422:
411:
410:
406:
402:
401:85.54.190.251
397:
396:
392:
388:
387:85.54.190.251
370:
366:
343:
339:
335:
312:
309:
306:
299:
291:
288:
272:
269:
266:
262:
241:
238:
235:
213:
207:
199:
196:
180:
177:
174:
170:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
3054:
3003:
2855:
2822:I'll ask at
2816:
2609:
2606:
2603:
2600:
2597:
2594:
2232:
2085:
1247:
1156:
1023:
915:
912:
909:
906:
903:
900:
897:
894:
893:
890:
887:
883:
882:
879:
875:
874:
871:
867:
859:
852:
850:
517:
514:
511:
475:
437:
417:
398:
161:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
3014:Convolution
2859:example of
2611:46.17.97.92
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
3074:Categories
2806:n}" /: -->
508:Summations
494:Mathmensch
421:MathHisSci
2813:WP:SOURCE
2828:Jowa fan
2783:n}": -->
479:Jowa fan
464:Jowa fan
2845:series.
139:on the
36:scale.
2791:: -->
3062:talk
3041:talk
3037:Taku
3026:talk
2986:talk
2960:talk
2956:Ozob
2832:talk
2752:and
2615:talk
858:and
498:talk
483:talk
468:talk
425:talk
405:talk
391:talk
131:Low
3076::
3064:)
3043:)
3028:)
2988:)
2980:.
2962:)
2896:−
2834:)
2800:n}
2686:⋯
2617:)
2537:⇒
2418:⇒
2253:⇒
2200:−
2182:∑
2150:−
2132:∑
2098:∑
1913:⇒
1748:⇒
1606:⇒
1487:⇒
1391:⇒
1318:⇒
1268:⇒
1228:−
1190:∑
1169:∑
977:∑
968:⋅
933:∑
818:−
791:∑
759:−
732:∑
710:−
692:∑
688:−
680:−
642:∑
618:∑
579:∑
570:⋅
535:∑
500:)
485:)
470:)
447:∑
427:)
407:)
393:)
336:∑
289:−
278:∞
263:∑
197:−
186:∞
171:∑
3060:(
3039:(
3024:(
2984:(
2958:(
2945:.
2931:1
2928:+
2925:n
2919:/
2913:1
2910:+
2907:n
2903:)
2899:1
2893:(
2890:=
2885:n
2881:b
2877:=
2872:n
2868:a
2856:n
2830:(
2808:.
2794:n
2788:k
2765:k
2761:b
2738:k
2734:a
2711:n
2707:c
2683:+
2678:2
2674:c
2670:+
2665:1
2661:c
2638:n
2634:c
2613:(
2578:4
2574:a
2568:2
2564:b
2560:+
2555:4
2551:b
2545:2
2541:a
2534:2
2531:=
2528:k
2505:5
2501:a
2495:1
2491:b
2487:+
2482:4
2478:a
2472:1
2468:b
2464:+
2459:5
2455:b
2449:1
2445:a
2441:+
2436:4
2432:b
2426:1
2422:a
2415:1
2412:=
2409:k
2386:6
2382:a
2376:0
2372:b
2368:+
2363:5
2359:a
2353:0
2349:b
2345:+
2340:4
2336:a
2330:0
2326:b
2322:+
2317:6
2313:b
2307:0
2303:a
2299:+
2294:5
2290:b
2284:0
2280:a
2276:+
2271:4
2267:b
2261:0
2257:a
2250:0
2247:=
2244:k
2218:)
2213:i
2209:a
2203:k
2197:6
2192:4
2189:=
2186:i
2176:k
2172:b
2168:+
2163:i
2159:b
2153:k
2147:6
2142:4
2139:=
2136:i
2126:k
2122:a
2118:(
2113:2
2108:0
2105:=
2102:k
2069:0
2065:b
2059:6
2055:a
2051:+
2046:1
2042:b
2036:5
2032:a
2028:+
2023:2
2019:b
2013:4
2009:a
2005:+
2000:3
1996:b
1990:3
1986:a
1982:+
1977:4
1973:b
1967:2
1963:a
1959:+
1954:5
1950:b
1944:1
1940:a
1936:+
1931:6
1927:b
1921:0
1917:a
1910:6
1907:=
1904:k
1881:0
1877:b
1871:5
1867:a
1863:+
1858:1
1854:b
1848:4
1844:a
1840:+
1835:2
1831:b
1825:3
1821:a
1817:+
1812:3
1808:b
1802:2
1798:a
1794:+
1789:4
1785:b
1779:1
1775:a
1771:+
1766:5
1762:b
1756:0
1752:a
1745:5
1742:=
1739:k
1716:0
1712:b
1706:4
1702:a
1698:+
1693:1
1689:b
1683:3
1679:a
1675:+
1670:2
1666:b
1660:2
1656:a
1652:+
1647:3
1643:b
1637:1
1633:a
1629:+
1624:4
1620:b
1614:0
1610:a
1603:4
1600:=
1597:k
1574:0
1570:b
1564:3
1560:a
1556:+
1551:1
1547:b
1541:2
1537:a
1533:+
1528:2
1524:b
1518:1
1514:a
1510:+
1505:3
1501:b
1495:0
1491:a
1484:3
1481:=
1478:k
1455:0
1451:b
1445:2
1441:a
1437:+
1432:1
1428:b
1422:1
1418:a
1414:+
1409:2
1405:b
1399:0
1395:a
1388:2
1385:=
1382:k
1359:0
1355:b
1349:1
1345:a
1341:+
1336:1
1332:b
1326:0
1322:a
1315:1
1312:=
1309:k
1286:0
1282:b
1276:0
1272:a
1265:0
1262:=
1259:k
1231:i
1225:k
1221:b
1215:i
1211:a
1205:k
1200:0
1197:=
1194:i
1184:6
1179:0
1176:=
1173:k
1142:)
1137:3
1133:b
1129:+
1124:2
1120:b
1116:+
1111:1
1107:b
1103:+
1098:0
1094:b
1090:(
1087:)
1082:3
1078:a
1074:+
1069:2
1065:a
1061:+
1056:1
1052:a
1048:+
1043:0
1039:a
1035:(
1008:)
1002:k
998:b
992:3
987:0
984:=
981:k
972:(
964:)
958:k
954:a
948:3
943:0
940:=
937:k
928:(
862:k
860:b
855:k
853:a
836:)
831:i
827:a
821:k
815:n
812:2
807:1
804:+
801:n
798:=
795:i
785:k
781:b
777:+
772:i
768:b
762:k
756:n
753:2
748:1
745:+
742:n
739:=
736:i
726:k
722:a
718:(
713:1
707:n
702:0
699:=
696:k
683:i
677:k
673:b
667:i
663:a
657:k
652:0
649:=
646:i
636:n
633:2
628:0
625:=
622:k
614:=
610:)
604:k
600:b
594:n
589:0
586:=
583:k
574:(
566:)
560:k
556:a
550:n
545:0
542:=
539:k
530:(
496:(
481:(
466:(
450:x
423:(
403:(
389:(
371:n
367:b
344:n
340:b
313:1
310:+
307:n
300:n
296:)
292:1
286:(
273:0
270:=
267:n
242:0
239:=
236:n
214:n
208:n
204:)
200:1
194:(
181:0
178:=
175:n
143:.
42::
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