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A217461 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(4-x)^(n-k).

%I A217461 #18 Jun 06 2013 17:06:10
%S A217461 1,4,16,72,350,1768,9120,47696,251974,1341560,7186912,38694000,
%T A217461 209187884,1134838736,6174666560,33681995936,184138474182,
%U A217461 1008642036184,5534504908640,30415064058160,167378028670788,922251663816368,5087347689155264,28091877168106592
%N A217461 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(4-x)^(n-k).
%C A217461 Radius of convergence of g.f. is r = (3-sqrt(7))/2 = 0.1771243444...
%C A217461 More generally, given
%C A217461 A(x) = Sum_{n>=1} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k),
%C A217461 then A(x) = 1 / sqrt( (1 - t*x + 2*x^2)^2 - 4*x^2 )
%C A217461 and the radius of convergence r satisfies: (1-r)^2 = r*(t-r) for t>0.
%H A217461 Vincenzo Librandi, <a href="/A217461/b217461.txt">Table of n, a(n) for n = 0..200</a>
%F A217461 G.f.: A(x) = 1 / sqrt( (1 - 4*x + 2*x^2)^2 - 4*x^2 ).
%F A217461 G.f.: A(x) = 1 / sqrt( (1-2*x+2*x^2)*(1-6*x+2*x^2) ).
%F A217461 G.f. satisfies: A(x) = [1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)] / (1-4*x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.
%F A217461 Recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 16*(n-1)*a(n-2) + 8*(2*n-3)*a(n-3) - 4*(n-2)*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F A217461 a(n) ~ sqrt(147+56*sqrt(7))*(3+sqrt(7))^n/(14*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 24 2012
%e A217461 G.f.: A(x) = 1 + 4*x + 16*x^2 + 72*x^3 + 350*x^4 + 1768*x^5 +...
%e A217461 where the g.f. equals the series:
%e A217461 A(x) = 1 +
%e A217461 x*((4-x) + x) +
%e A217461 x^2*((4-x)^2 + 2^2*x*(4-x) + x^2) +
%e A217461 x^3*((4-x)^3 + 3^2*x*(4-x)^2 + 3^2*x^2*(4-x) + x^3) +
%e A217461 x^4*((4-x)^4 + 4^2*x*(4-x)^3 + 6^2*x^2*(4-x)^2 + 4^2*x^3*(4-x) + x^4) +
%e A217461 x^5*((4-x)^5 + 5^2*x*(4-x)^4 + 10^2*x^2*(4-x)^3 + 10^2*x^3*(4-x)^2 + 5^2*x^4*(4-x) + x^5) +...
%t A217461 CoefficientList[Series[1/Sqrt[(1-2*x+2*x^2)*(1-6*x+2*x^2)], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%o A217461 (PARI) {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(4-x)^(m-k) + x*O(x^n))), n)}
%o A217461 for(n=0, 40, print1(a(n), ", "))
%Y A217461 Cf. A217615, A217616, A217617.
%K A217461 nonn
%O A217461 0,2
%A A217461 _Paul D. Hanna_, Oct 10 2012