A218692 Sum_{k=0..n} C(n,k)^6*C(n+k,k)^3.

1, 9, 1945, 783657, 333935001, 216152253009, 148273286805001, 112444816742316585, 93273051852487532953, 80885382627785790555009, 73726153308964013326434945, 69714999360408389332640853105, 67921574835559806028030517001225, 67965584346796032477336615843457665

Offset: 0

Author

Vaclav Kotesovec, Nov 04 2012

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Crossrefs

Cf. A005258, A218690.

Programs

Mathematica

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

Formula

a(n) ~ ((1+sqrt(5))/2)^(3*(5*n+4)-3/2)/(5^(1/4)*(2*Pi*n)^4*sqrt(3))

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)).