%I #14 Jun 07 2021 01:10:48
%S 1,10,236,8472,359944,16722896,822334816,42068907200,2215884717400,
%T 119364801362800,6545334930678816,364137834051739200,
%U 20502307365808906816,1166063313963833813632,66893439680369963627264
%N a(n) = 2^n * Sum_{k=0..n} binomial(2*k,k)^3 / 2^k.
%C The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)/B^k gives A006134 for B=1, A082590 (B=2), A132310 (B=3), A002457 (B=4), A144635 (B=5), A167713 (B=16).
%C The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).
%H Vincenzo Librandi, <a href="/A167867/b167867.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = 2^n * Sum_{k=0..n} binomial(2*k,k)^3 / 2^k.
%F Recurrence: n^3*a(n) = 2*(33*n^3 - 48*n^2 + 24*n - 4)*a(n-1) - 16*(2*n-1)^3*a(n-2). - _Vaclav Kotesovec_, Aug 13 2013
%F a(n) ~ 2^(6*n+5)/(31*(Pi*n)^(3/2)). - _Vaclav Kotesovec_, Aug 13 2013
%t Table[2^n Sum[Binomial[2k,k]^3/2^k,{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Mar 26 2012 *)
%Y Cf. A079727, A167867, A167868, A167869, A167870, A167872.
%Y Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859.
%K nonn
%O 0,2
%A _Alexander Adamchuk_, Nov 14 2009
%E More terms from _Sean A. Irvine_, Apr 27 2010
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