A216636 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 5^k.

1, 6, 66, 936, 14346, 231876, 3885456, 66767616, 1169068986, 20769386796, 373277526876, 6772297456656, 123834925330416, 2279408745325536, 42194656181618496, 784905308800229376, 14663340953943086106, 274968958499402854716, 5173516852494573136836

Offset: 0

Author

Vaclav Kotesovec, Sep 11 2012

Keywords

• nonn

Comments

Diagonal of rational function 1/(1 + y + z + x*y + y*z + 5*x*z + 6*x*y*z). - Gheorghe Coserea, Jul 01 2018

Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - 5*x*y*z). - Seiichi Manyama, Jul 11 2020

Links

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

Crossrefs

Cf. A206178, A206180, A216483.

Programs

Mathematica

Table[Sum[Binomial[n,k]^3*5^k,{k,0,n}],{n,0,20}]

PARI

a(n) = sum(k=0, n, binomial(n,k)^3 * 5^k); \\ Gheorghe Coserea, Jul 01 2018

Sage

A216636 = lambda n: hypergeometric([-n,-n,-n],[1,1], -5)
[Integer(A216636(n).n(100)) for n in (0..18)] # Peter Luschny, Sep 23 2014

Formula

Recurrence: (n+3)^2*(3*n+4)*a(n+3) = 6*(9*n^3+57*n^2+116*n+74)*a(n+2) + 3*(27*n^3+144*n^2+261*n+160)*a(n+1) + 216*(3*n+7)*(n+1)^2*a(n).

a(n) ~ (1 + 5^(1/3))^(3*n+2) / (2*sqrt(3)*5^(1/3)*Pi*n). - Vaclav Kotesovec, Sep 19 2012, simplified Apr 24 2025

G.f.: hypergeom([1/3, 2/3],[1],5*27*x^2/(1-6*x)^3)/(1-6*x). - Mark van Hoeij, May 02 2013

a(n) = hypergeom([-n,-n,-n],[1,1], -5). - Peter Luschny, Sep 23 2014

G.f. y=A(x) satisfies: x*(3*x + 1)*(216*x^3 + 27*x^2 + 18*x - 1)*y'' + (1944*x^4 + 1026*x^3 + 135*x^2 + 36*x - 1)*y' + 6*(108*x^3 + 69*x^2 + 2*x + 1)*y. - Gheorghe Coserea, Jul 01 2018

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014