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Sum_{k=0..n} C(n,k)^6*C(n+k,k)^3.
8

%I #11 Sep 12 2015 11:00:29

%S 1,9,1945,783657,333935001,216152253009,148273286805001,

%T 112444816742316585,93273051852487532953,80885382627785790555009,

%U 73726153308964013326434945,69714999360408389332640853105,67921574835559806028030517001225,67965584346796032477336615843457665

%N Sum_{k=0..n} C(n,k)^6*C(n+k,k)^3.

%H Vincenzo Librandi, <a href="/A218692/b218692.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(5))/2)^(3*(5*n+4)-3/2)/(5^(1/4)*(2*Pi*n)^4*sqrt(3))

%F Generally, Sum_{k=0..n} C(n,k)^(2*q)*C(n+k,k)^q is asymptotic to ((1+sqrt(5))/2)^(q*(5*n+4)-3/2)/(5^(1/4)*sqrt(q*(2*Pi*n)^(3*q-1))) * (1-(25*q^2+96*q-61)/(120*q*n)-(13*q^2-36*q+17)*sqrt(5)/(60*q*n)).

%t Table[Sum[Binomial[n,k]^6*Binomial[n+k,k]^3,{k,0,n}],{n,0,20}]

%Y Cf. A005258, A218690.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 04 2012