327:
45:
322:{\displaystyle (x+m+1)\sum _{i=0}^{m}(-1)^{i}{\dbinom {x+y+i}{m-i}}{\dbinom {y+2i}{i}}-\sum _{i=0}^{m}{\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\dbinom {x}{m}}.}
596:
548:
601:
363:
337:
After Sun's publication of this identity in 2002, five other proofs were obtained by various mathematicians:
28:
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32:
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404:
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Identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002
546:
Sun, Zhi-Wei (2008), "On sums of binomial coefficients and their applications",
36:
571:
356:
562:
409:
374:
532:
INTEGERS: The
Electronic Journal of Combinatorial Number Theory
509:
INTEGERS: The
Electronic Journal of Combinatorial Number Theory
486:
INTEGERS: The
Electronic Journal of Combinatorial Number Theory
463:
INTEGERS: The
Electronic Journal of Combinatorial Number Theory
437:
INTEGERS: The
Electronic Journal of Combinatorial Number Theory
397:
INTEGERS: The
Electronic Journal of Combinatorial Number Theory
293:
216:
157:
111:
48:
525:"A curious identity involving binomial coefficients"
502:"A generating functions proof of a curious identity"
390:"A combinatorial proof of Sun's 'curious' identity"
321:
309:
296:
248:
219:
184:
160:
149:
114:
479:"A Riordan array proof of a curious identity"
430:"Jensen proof of a curious binomial identity"
8:
561:
408:
362:Chu and Claudio's proof with the help of
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85:
74:
47:
500:Panholzer, A.; Prodinger, H. (2002),
7:
456:"A WZ proof of a 'curious' identity"
348:Merlini and Sprugnoli's proof using
341:Panholzer and Prodinger's proof via
477:Merlini, D.; Sprugnoli, R. (2002),
355:Ekhad and Mohammed's proof by the
300:
223:
164:
118:
14:
428:Chu, W.; Claudio, L.V.D. (2003),
289:
277:
265:
255:
101:
91:
67:
49:
1:
597:Factorial and binomial topics
618:
572:10.1016/j.disc.2007.08.046
454:; Mohammed, M. (2003),
35:, first established by
323:
214:
90:
25:Sun's curious identity
523:Sun, Zhi-Wei (2002),
324:
194:
70:
33:binomial coefficients
602:Algebraic identities
549:Discrete Mathematics
343:generating functions
46:
419:2004math......1216C
388:Callan, D. (2004),
371:combinatorial proof
319:
314:
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556:(18): 4231–4245,
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27:is the following
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89:
84:
617:
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586:
563:math.NT/0404385
545:
527:
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504:
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458:
450:
432:
427:
410:math.CO/0401216
392:
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165:
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136:
119:
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100:
44:
43:
17:
12:
11:
5:
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584:
542:
541:
519:
518:
496:
495:
473:
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424:
423:
383:
380:
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378:
377:and colorings.
367:
360:
353:
350:Riordan arrays
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51:
15:
13:
10:
9:
6:
4:
3:
2:
614:
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581:
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569:
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457:
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78:
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42:
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34:
30:
26:
22:
21:combinatorics
553:
547:
535:
531:
512:
508:
489:
485:
466:
462:
452:Ekhad, S. B.
440:
436:
400:
396:
336:
24:
18:
366:'s formula;
37:Zhi-Wei Sun
591:Categories
382:References
373:involving
31:involving
369:Callan's
284:−
259:−
240:−
196:∑
192:−
141:−
95:−
72:∑
39:in 2002:
580:14089498
29:identity
415:Bibcode
403:: A05,
375:dominos
359:method;
578:
364:Jensen
333:Proofs
576:S2CID
558:arXiv
538:: A04
528:(PDF)
515:: A06
505:(PDF)
492:: A08
482:(PDF)
469:: A06
459:(PDF)
443:: A20
433:(PDF)
405:arXiv
393:(PDF)
568:doi
554:308
19:In
593::
574:,
566:,
552:,
534:,
530:,
511:,
507:,
488:,
484:,
465:,
461:,
439:,
435:,
413:,
399:,
395:,
357:WZ
23:,
583:.
570::
560::
540:.
536:2
517:.
513:2
494:.
490:2
471:.
467:3
445:.
441:3
422:.
417::
407::
401:4
352:;
345:;
317:.
310:)
305:m
302:x
297:(
290:)
287:m
281:x
278:(
275:=
270:i
266:)
262:4
256:(
249:)
243:i
237:m
232:i
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226:x
220:(
211:m
206:0
203:=
200:i
185:)
180:i
176:i
173:2
170:+
167:y
161:(
150:)
144:i
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127:y
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98:1
92:(
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82:0
79:=
76:i
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65:1
62:+
59:m
56:+
53:x
50:(
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