A219564 Sum(binomial(n+k,k)^6, k=0..n).

1, 65, 47386, 65004097, 119498671876, 260128695981674, 632156164654144530, 1659900189891175027265, 4616088190888638302435080, 13418259230056806455830305940, 40401802613222456104862752944356, 125182282922559710456869140648653290, 397195659937314116991934285462527257236

Offset: 0

Author

Vaclav Kotesovec, Nov 23 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. A001700 (q=1), A112029 (q=2), A112028 (q=3), A219562 (q=4), A219563 (q=5).

Programs

Mathematica

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

Formula

a(n) ~ 2^(12*n+6)/(63*Pi^3*n^3)

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))