The On-Line Encyclopedia of Integer Sequences (OEIS)

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A188912

Binomial convolution of the binomial coefficients bin(3n,n)/(2n+1) (A001764).
(history; published version)

#14 by N. J. A. Sloane at Wed Aug 24 09:28:07 EDT 2022

STATUS

proposed

approved

#13 by Jon E. Schoenfield at Wed Aug 24 01:20:20 EDT 2022

STATUS

editing

proposed

#12 by Jon E. Schoenfield at Wed Aug 24 01:18:17 EDT 2022

FORMULA

a(n) = sum(Sum_{k=0..n} binomial(n, k)*binomial(3*k, k)/(2*k+1)*binomial(3*n-3*k, n-k)/((2*k+1)*(2*n-2*k+1), k=0..n).

E.g.f.: F(1/3,2/3;1,3/2;27*x/4)^2, where F(a1,a2;b1,b2;z) is a hypergeometric series.

STATUS

approved

editing

Discussion

Wed Aug 24

01:20

Jon E. Schoenfield: Rearrangement of 1st Formula entry (because of expression of the form A*B/C*D/E) okay?

#11 by Vaclav Kotesovec at Mon Jun 10 04:50:40 EDT 2019

STATUS

editing

approved

#10 by Vaclav Kotesovec at Mon Jun 10 04:50:18 EDT 2019

FORMULA

From Vaclav Kotesovec, Jun 10 2019: (Start)

Recurrence: 8*n^2*(n+1)*(2*n+1)^2*(9*n^3-54*n^2+84*n-35)*a(n) = 24*n*(324*n^7-2187*n^6+4689*n^5-4185*n^4+1464*n^3+122*n^2-223*n+44)*a(n-1) - 18*(n-1)*(3645*n^7-30618*n^6+96066*n^5-144585*n^4+103662*n^3-21834*n^2-10860*n+4480)*a(n-2) + 2187*(n-2)^2*(n-1)*(3*n-7)*(3*n-5)*(9*n^3-27*n^2+3*n+4)*a(n-3).

a(n) ~ 3^(3*n + 1) / (Pi * n^3 * 2^(n + 1)). (End)

STATUS

approved

editing

#9 by Russ Cox at Fri Mar 30 18:55:30 EDT 2012

AUTHOR

_Emanuele Munarini (emanuele.munarini(AT)polimi.it), _, Apr 13 2011

Discussion

Fri Mar 30

18:55

OEIS Server: https://oeis.org/edit/global/279

#8 by Joerg Arndt at Tue May 24 06:31:37 EDT 2011

STATUS

proposed

approved

#7 by Vincenzo Librandi at Tue May 24 01:21:06 EDT 2011

LINKS

Vincenzo Librandi, <a href="/A188912/b188912.txt">Table of n, a(n) for n = 0..93</a>

STATUS

approved

proposed

#6 by Joerg Arndt at Thu Apr 14 03:53:33 EDT 2011

STATUS

proposed

approved

#5 by Joerg Arndt at Thu Apr 14 03:53:28 EDT 2011

KEYWORD

nonn,easy,changed