A216698 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 6^k.

1, 7, 85, 1351, 23281, 422527, 7951069, 153458935, 3018043777, 60225528727, 1215821974885, 24777776573095, 508935634491025, 10522995625652335, 218814097786515085, 4572338217781407031, 95953172529722919937, 2021236451413828339495, 42719661851354642952181

Offset: 0

Author

Vaclav Kotesovec, Sep 15 2012

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• V. Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012

Crossrefs

Cf. A000172 (x=1), A206178 (x=2), A206180 (x=3), A216483 (x=4), A216636 (x=5), A216696.

Programs

Mathematica

Table[Sum[Binomial[n, k]^3*6^k, {k, 0, n}], {n, 0, 25}]

Sage

A216698 = lambda n: hypergeometric([-n,-n,-n], [1,1], -6)
[Integer(A216698(n).n(100)) for n in (0..18)] # Peter Luschny, Sep 23 2014

Formula

General recurrecnce for Sum_{k=0..n} binomial(n,k)^3*x^k (this is case x=6): (n+3)^2*(3*n+4)*a(n+3) -(9*n^3+57*n^2+116*n+74)*(x+1)*a(n+2) +(3*n+5)*(3*n^2*(x^2-7*x+1)+11*n*(x^2-7*x+1)+9*x^2-66*x+9)*a(n+1) -(n+1)^2*(3*n+7)*(x+1)^3*a(n) = 0.

a(n) ~ (1+6^(1/3))^2/(2*2^(1/3)*3^(5/6)*Pi) * (1+6^(1/3))^(3*n)/n. - Vaclav Kotesovec, Sep 19 2012

G.f.: hypergeom([1/3, 2/3],[1],6*27*x^2/(1-7*x)^3)/(1-7*x). - Mark van Hoeij, May 02 2013

a(n) = hypergeom([-n,-n,-n],[1,1], -6). - Peter Luschny, Sep 23 2014

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014