A218689 - OEIS

%I #11 Sep 12 2015 11:00:28

%S 1,65,93313,795985985,8178690000001,93706344780048065,

%T 1453730786373283012225,26552497154713885161031745,

%U 513912636558068387176582890625,10769375530849394271690330588432065,243282405272735566295972089793676717313,5763401688773271719278313934033057270226625

%N Sum_{k=0..n} C(n,k)^6*C(n+k,k)^6.

%H Vincenzo Librandi, <a href="/A218689/b218689.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) ~ (1+sqrt(2))^(6*(2n+1))/(2^(17/4)*(Pi*n)^(11/2)*sqrt(3))

%F Generally, Sum_{k=0..n} C(n,k)^p*C(n+k,k)^p is asymptotic to (1+sqrt(2))^(p*(2*n+1))/(2^(p/2+3/4)*(Pi*n)^(p-1/2)*sqrt(p)) * (1-(2*p-1)/(4*n)+(4*p^2+24*p-19)*sqrt(2)/(96*p*n))

%t Table[Sum[Binomial[n,k]^6*Binomial[n+k,k]^6,{k,0,n}],{n,0,20}]

%Y Cf. A001850 (p=1), A005259 (p=2), A092813 (p=3), A092814 (p=4), A092815 (p=5).

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 04 2012