A112019 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k)^2.

1, 5, 55, 749, 11251, 178835, 2949115, 49906925, 860905315, 15071939255, 266982872905, 4774722189275, 86070844191775, 1561948324845095, 28507384046515555, 522867506128197869, 9631571375362268515, 178094411589895650815, 3304192479145474141741, 61487420580006795749999

Offset: 0

Author

N. J. A. Sloane, Nov 28 2005

Keywords

• nonn

Comments

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

Diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - z). - Seiichi Manyama, Apr 30 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
• V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Crossrefs

Cf. A218693, A111968, A014178, A014180, A218689, A218692.

Cf. A383536, A383537, A383538.

Programs

Maple

seq(add((multinomial(n+k,n-k,k,k))*binomial(n+k,k),k=0..n),n=0..19); # Zerinvary Lajos, Oct 18 2006
ogf := hypergeom([1/12,5/12],[1], -1728*(x^3+5*x^2+39*x-2)*x^4 / (x^4+4*x^3+30*x^2-20*x+1)^3 ) / (x^4+4*x^3+30*x^2-20*x+1)^(1/4);
series(ogf, x=0, 30); # Mark van Hoeij, Jan 22 2013

Mathematica

Table[HypergeometricPFQ[{-n, 1 + n, 1 + n}, {1, 1}, -1], {n, 0, 20}] (* Olivier Gérard, Apr 23 2009 *)
Table[Sum[Binomial[n,k]*Binomial[n+k,k]^2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Nov 04 2012 *)

PARI

a(n) = sum(k=0, n, binomial(n,k)*binomial(n+k,k)^2); \\ Michel Marcus, Mar 09 2016

Formula

a(n) = 3F2( {-n, 1 + n, 1 + n} ; {1, 1} )(-1). - Olivier Gérard, Apr 23 2009

a(n) ~ (1+r)^(4*n+5/2)/r^(3*n+5/2)/(2*Pi*n)*sqrt((1-r)/(3-r)), where r is positive real root of the equation (1-r)*(1+r)^2=r^3, r = 1/6*((44-3*sqrt(177))^(1/3)+(44+3*sqrt(177))^(1/3)-1) = 0.82948354095849... - Vaclav Kotesovec, Nov 04 2012

Recurrence: 2*n^2*(59*n - 83)*a(n) = (2301*n^3 - 5538*n^2 + 3797*n - 800)*a(n-1) + 5*(59*n^3 - 201*n^2 + 213*n - 64)*a(n-2) + (59*n - 24)*(n-2)^2*a(n-3). - Vaclav Kotesovec, Nov 04 2012

G.f. y=A(x) satisfies: 0 = x*(5*x + 8)*(x^3 + 5*x^2 + 39*x - 2)*y'' + (15*x^4 + 82*x^3 + 315*x^2 + 624*x - 16)*y' + (5*x^3 + 21*x^2 + 80)*y. - Gheorghe Coserea, Jul 01 2018