A241756 A finite sum of products of binomial coefficients: Sum_(m=0..n) binomial(-1/4, m)^2*binomial(-1/4, n-m)^2 (C. C. Grosjean's problem, denominators).

1, 8, 512, 4096, 2097152, 16777216, 1073741824, 8589934592, 35184372088832, 281474976710656, 18014398509481984, 144115188075855872, 73786976294838206464, 590295810358705651712, 37778931862957161709568, 302231454903657293676544

Offset

0,2

Comments

This sequence seems to appear also as denominators of A277232, A277234, and A278143. - Wolfdieter Lang, Nov 16 2016

References

E. S. Andersen and M. E. Larsen. A finite sum of products of binomial coefficients, Problem 92-18, by C. C. Grosjean, Solution. SIAM Rev. 35 (1993), 645-646.

Links

Table of n, a(n) for n=0..15.

P. Flajolet, B. Salvy, and Helmut Prodinger, A Finite Sum of Products of Binomial Coefficients, Problem 92-18 by C. C. Grosjean, Solution. SIAM Rev. 35 (1993), 645-646.

C. C. Grosjean, Problem no. 92-18, SIAM Rev. 34 (1992), p. 649.

M. E. Larsen, Summa Summarum, page 114.

Formula

GAMMA(3/4)^2 * 4F3(1/4, 1/4, -n, -n; 1, 3/4-n, 3/4-n; 1)/(GAMMA(3/4-n)^2*GAMMA(n+1)^2).

binomial(2n, n)^2*binomial(n-1/2, 2n)*(-1/4)^n.

Conjecture (from sequencedb.net): a(n) = 8^A005187(n). - R. J. Mathar, Jun 30 2021

Mathematica

a[n_] := Binomial[2*n, n]^2*Binomial[n-1/2, 2*n]*(-1/4)^n; Table[a[n]//Denominator, {n, 0, 20}]

Crossrefs

Cf. A241755 (numerators), A277232, A277234, A278143.

Sequence in context: A236077 A061460 A016935 * A236220 A128794 A195804

Adjacent sequences: A241753 A241754 A241755 * A241757 A241758 A241759

Keyword

nonn,frac

Author

Jean-François Alcover, Apr 28 2014

Status

approved