A074635 a(0)=1, a(n) = Sum_{k=0..n} (binomial(n,k)^2 * binomial(n+k,k+1)^2)/n^2.

1, 2, 14, 162, 2422, 42366, 824210, 17315570, 385569110, 8985969258, 217250577766, 5413255505686, 138331193497122, 3611539459764086, 96043941386294106, 2595400550031689938, 71127992129228040790, 1973658406181654681730, 55374306085337133154190

Offset: 0

Author

Karol A. Penson, Aug 26 2002

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..659

Crossrefs

Cf. A006318.

Programs

Maple

sq := (x^2-34*x+1)^(1/2);
f := 54*(sq+x-1)^2/(sq+x+7)^3;
H1 := hypergeom([1/3,-1/3],[1],f);
H2 := hypergeom([1/3, 2/3],[1],f);
ogf := (2*sq-6*x+6)*H1^2/(9*x) + (4*x-8-2*sq)*H1*H2/(9*x) + (sq^3-5*x^3-105*x^2+129*x+13)*H2^2/(36*x*(x+1)^2) - (x+1)/(6*x);
series(ogf,x=0,20);  # Mark van Hoeij, Apr 04 2013

Mathematica

Join[{1}, Table[Sum[Binomial[n, k]^2 * Binomial[n + k, k + 1]^2, {k, 0, n}]/n^2, {n, 25}]] (* T. D. Noe, Apr 05 2013 *)
Table[HypergeometricPFQ[{-n, -n, n+1, n+1}, {1, 2, 2}, 1], {n, 0, 18}] (* Jean-François Alcover, Nov 06 2016 *)

PARI

a(n)=my(t=n); if(!n, return(1)); sum(k=1,n, t*=(n-k+1)*(n+k)/k/(k+1); t^2)/n^2+1 \\ Charles R Greathouse IV, Nov 07 2016

Formula

Special values of the hypergeometric function 4F3, in Maple notation: a(n)= hypergeom([n+1, n+1, -n, -n], [1, 2, 2], 1), n=0, 1...

Recurrence: (n+1)^3*(3*n^2-6*n+2)*a(n) = (2*n-1)*(51*n^4 - 102*n^3 + 19*n^2 + 32*n - 14)*a(n-1) - (3*n^2-1)*(n-2)^3*a(n-2). - Vaclav Kotesovec, Jun 29 2013

a(n) ~ sqrt(48+34*sqrt(2))*(17+12*sqrt(2))^n/(4*n^(7/2)*Pi^(3/2)). - Vaclav Kotesovec, Jun 29 2013

Extensions

An initial 1 deleted by Mark van Hoeij, Apr 04 2013