%I #31 Nov 07 2016 10:51:53
%S 1,2,14,162,2422,42366,824210,17315570,385569110,8985969258,
%T 217250577766,5413255505686,138331193497122,3611539459764086,
%U 96043941386294106,2595400550031689938,71127992129228040790,1973658406181654681730,55374306085337133154190
%N a(0)=1, a(n) = Sum_{k=0..n} (binomial(n,k)^2 * binomial(n+k,k+1)^2)/n^2.
%H Seiichi Manyama, <a href="/A074635/b074635.txt">Table of n, a(n) for n = 0..659</a>
%F 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...
%F 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
%F a(n) ~ sqrt(48+34*sqrt(2))*(17+12*sqrt(2))^n/(4*n^(7/2)*Pi^(3/2)). - _Vaclav Kotesovec_, Jun 29 2013
%p sq := (x^2-34*x+1)^(1/2);
%p f := 54*(sq+x-1)^2/(sq+x+7)^3;
%p H1 := hypergeom([1/3,-1/3],[1],f);
%p H2 := hypergeom([1/3, 2/3],[1],f);
%p 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);
%p series(ogf,x=0,20); # _Mark van Hoeij_, Apr 04 2013
%t 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 *)
%t Table[HypergeometricPFQ[{-n, -n, n+1, n+1}, {1, 2, 2}, 1], {n, 0, 18}] (* _Jean-François Alcover_, Nov 06 2016 *)
%o (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
%Y Cf. A006318.
%K nonn
%O 0,2
%A _Karol A. Penson_, Aug 26 2002
%E An initial 1 deleted by _Mark van Hoeij_, Apr 04 2013