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%I #21 Jun 30 2021 11:39:42
%S 1,8,512,4096,2097152,16777216,1073741824,8589934592,35184372088832,
%T 281474976710656,18014398509481984,144115188075855872,
%U 73786976294838206464,590295810358705651712,37778931862957161709568,302231454903657293676544
%N 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).
%C This sequence seems to appear also as denominators of A277232, A277234, and A278143. - _Wolfdieter Lang_, Nov 16 2016
%D 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.
%H P. Flajolet, B. Salvy, and Helmut Prodinger, <a href="http://dx.doi.org/10.1137/1035147">A Finite Sum of Products of Binomial Coefficients</a>, Problem 92-18 by C. C. Grosjean, Solution. SIAM Rev. 35 (1993), 645-646.
%H C. C. Grosjean, <a href="http://dx.doi.org/10.1137/1034122">Problem no. 92-18</a>, SIAM Rev. 34 (1992), p. 649.
%H M. E. Larsen, <a href="http://www.math.ku.dk/~mel/larsen.pdf">Summa Summarum</a>, page 114.
%F 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).
%F binomial(2n, n)^2*binomial(n-1/2, 2n)*(-1/4)^n.
%F Conjecture (from sequencedb.net): a(n) = 8^A005187(n). - _R. J. Mathar_, Jun 30 2021
%t a[n_] := Binomial[2*n, n]^2*Binomial[n-1/2, 2*n]*(-1/4)^n; Table[a[n]//Denominator, {n, 0, 20}]
%Y Cf. A241755 (numerators), A277232, A277234, A278143.
%K nonn,frac
%O 0,2
%A _Jean-François Alcover_, Apr 28 2014
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