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%I #10 Sep 12 2015 11:00:29
%S 1,65,47386,65004097,119498671876,260128695981674,632156164654144530,
%T 1659900189891175027265,4616088190888638302435080,
%U 13418259230056806455830305940,40401802613222456104862752944356,125182282922559710456869140648653290,397195659937314116991934285462527257236
%N Sum(binomial(n+k,k)^6, k=0..n).
%H Vincenzo Librandi, <a href="/A219564/b219564.txt">Table of n, a(n) for n = 0..200</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of generalized Apery sequences with powers of binomial coefficients</a>, Nov 04 2012
%F a(n) ~ 2^(12*n+6)/(63*Pi^3*n^3)
%F Generally (for q > 0), Sum_{k=0..n} C(n + k,k)^q is asymptotic to 2^((2*n+1)*q)/((2^q-1)*(Pi*n)^(q/2)) * (1 - q/(2*n)*(1/4+1/(2^q-1)^2) + O(1/n^2))
%t Table[Sum[Binomial[n+k,k]^6, {k,0,n}], {n,0,20}]
%Y Cf. A001700 (q=1), A112029 (q=2), A112028 (q=3), A219562 (q=4), A219563 (q=5).
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Nov 23 2012
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