A218689 Sum_{k=0..n} C(n,k)^6*C(n+k,k)^6.

1, 65, 93313, 795985985, 8178690000001, 93706344780048065, 1453730786373283012225, 26552497154713885161031745, 513912636558068387176582890625, 10769375530849394271690330588432065, 243282405272735566295972089793676717313, 5763401688773271719278313934033057270226625

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

Programs

Mathematica

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

Formula

a(n) ~ (1+sqrt(2))^(6*(2n+1))/(2^(17/4)*(Pi*n)^(11/2)*sqrt(3))

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))