%I #4 Oct 10 2012 04:18:22
%S 1,1,2,7,23,72,227,725,2332,7537,24465,79720,260633,854539,2808768,
%T 9252315,30536925,100959558,334301159,1108483583,3680134756,
%U 12231824111,40697552035,135536687436,451776392011,1507088458381,5031254413136,16807872970501,56185887793379
%N G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-2*x)^k.
%C Radius of convergence of g.f. A(x) is |x| < 0.29392962790...
%C More generally, given
%C A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-t*x)^k then
%C A(x) = (1-t*x) / sqrt( (1-(t+1)*x)^2*(1+x^2) + (2*t-3)*x^2 - 2*t*(t-1)*x^3 ).
%F G.f.: (1-2*x) / sqrt(1 - 6*x + 11*x^2 - 10*x^3 + 9*x^4).
%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 23*x^4 + 72*x^5 + 227*x^6 + 725*x^7 +...
%e where the g.f. equals the series:
%e A(x) = 1 +
%e x*(1 + x/(1-2*x)) +
%e x^2*(1 + 2^2*x/(1-2*x) + x^2/(1-2*x)^2) +
%e x^3*(1 + 3^2*x/(1-2*x) + 3^2*x^2/(1-2*x)^2 + x^3/(1-2*x)^3) +
%e x^4*(1 + 4^2*x/(1-2*x) + 6^2*x^2/(1-2*x)^2 + 4^2*x^3/(1-2*x)^3 + x^4/(1-2*x)^4) +
%e x^5*(1 + 5^2*x/(1-2*x) + 10^2*x^2/(1-2*x)^2 + 10^2*x^3/(1-2*x)^3 + 5^2*x^4/(1-2*x)^4 + x^5/(1-2*x)^5) +...
%o (PARI) {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k/(1-2*x +x*O(x^n))^k )), n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A217661, A217665, A217666.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 10 2012