cp's OEIS Frontend

A187638 A convolution of binomial coefficients.

%I A187638 #16 Apr 06 2014 05:35:00
%S A187638 1,1,1,8,77,824,9448,113728,1419349,18215576,239034104,3194168768,
%T A187638 43331889928,595375647424,8270102936896,115966112244224,
%U A187638 1639592525879333,23350785267093848,334714889445506584,4825695011761675456,69936530485677013528
%N A187638 A convolution of binomial coefficients.
%H A187638 Vincenzo Librandi, <a href="/A187638/b187638.txt">Table of n, a(n) for n = 0..200</a>
%F A187638 G.f.: 9/8 - (1/8)*(K(16*x)/Pi)^2, where K(x) is the elliptic integral of the first kind (as defined in Mathematica).
%F A187638 a(n) = (1/8)*sum(C(2k,k)^2/(2k-1)*C(2n-2k,n-k)^2/(2n-2k-1), k=0..n) for n >= 1.
%F A187638 Recurrence: n^3*a(n) = 8*(4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - 256*(n-3)*(n-2)*(n-1)*a(n-2). - _Vaclav Kotesovec_, Apr 06 2014
%F A187638 a(n) ~ 4^(2*n-1)/(Pi^2*n^2). - _Vaclav Kotesovec_, Apr 06 2014
%p A187638 seq((-1/8)*add(binomial(2*k,k)^2/(2*k-1)*binomial(2*(n-k), n-k)^2/(2*(n-k)-1), k=0..n),n=1..12);
%t A187638 Table[-1/8 Sum[Binomial[2k,k]^2/(2k-1) Binomial[2n-2k,n-k]^2/(2n-2k-1), {k, 0, n}], {n, 1, 20}]
%o A187638 (Maxima) makelist((-1/8)*sum(binomial(2*k,k)^2/(2*k-1)*binomial(2*(n-k),n-k)^2/(2*(n-k)-1),k,0,n),n,1,12);
%Y A187638 Cf. A057718.
%K A187638 nonn,easy
%O A187638 0,4
%A A187638 _Emanuele Munarini_, Mar 12 2011