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

0,2

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.

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).

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

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

Mathematica

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

Prog

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

Crossrefs

Cf. A206178, A206180, A216483.

Sequence in context: A151832 A133306 A371681 * A169715 A211824 A360976

Adjacent sequences: A216633 A216634 A216635 * A216637 A216638 A216639

Keyword

nonn

Author

Vaclav Kotesovec, Sep 11 2012

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

Status

approved