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A217617 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(3-x)^(n-k).

%I A217617 #8 Sep 16 2013 12:46:29
%S A217617 1,3,9,33,133,549,2295,9711,41505,178749,774387,3370995,14733043,
%T A217617 64608555,284143257,1252749777,5535201733,24503713893,108659076723,
%U A217617 482566381299,2146042722591,9555487997247,42594294578949,190060286569677,848858809506279,3794468370955587
%N A217617 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(3-x)^(n-k).
%C A217617 Radius of convergence of g.f. is r = (5-sqrt(17))/4 = 0.21922359...
%C A217617 More generally, given
%C A217617 A(x) = Sum_{n>=1} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k),
%C A217617 then A(x) = 1 / sqrt( (1 - t*x + 2*x^2)^2 - 4*x^2 )
%C A217617 and the radius of convergence r satisfies: (1-r)^2 = r*(t-r) for t>0.
%F A217617 G.f.: A(x) = 1 / sqrt( (1 - 3*x + 2*x^2)^2 - 4*x^2 ).
%F A217617 G.f.: A(x) = 1 / sqrt( (1-x+2*x^2)*(1-5*x+2*x^2) ).
%F A217617 G.f. satisfies: A(x) = [1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)] / (1-3*x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan
%F A217617 numbers.
%F A217617 Recurrence: n*a(n) = 3*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2) + 6*(2*n-3)*a(n-3) - 4*(n-2)*a(n-4). - _Vaclav Kotesovec_, Sep 16 2013
%F A217617 a(n) ~ 2*((5+sqrt(17))/2)^n/sqrt((42*sqrt(17)-170)*Pi*n). - _Vaclav Kotesovec_, Sep 16 2013
%e A217617 G.f.: A(x) = 1 + 3*x + 9*x^2 + 33*x^3 + 133*x^4 + 549*x^5 + 2295*x^6 +...
%e A217617 where the g.f. equals the series:
%e A217617 A(x) = 1 +
%e A217617 x*((3-x) + x) +
%e A217617 x^2*((3-x)^2 + 2^2*x*(3-x) + x^2) +
%e A217617 x^3*((3-x)^3 + 3^2*x*(3-x)^2 + 3^2*x^2*(3-x) + x^3) +
%e A217617 x^4*((3-x)^4 + 4^2*x*(3-x)^3 + 6^2*x^2*(3-x)^2 + 4^2*x^3*(3-x) + x^4) +
%e A217617 x^5*((3-x)^5 + 5^2*x*(3-x)^4 + 10^2*x^2*(3-x)^3 + 10^2*x^3*(3-x)^2 + 5^2*x^4*(3-x) + x^5) +...
%t A217617 CoefficientList[Series[1/Sqrt[(1-3*x+2*x^2)^2-4*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 16 2013 *)
%o A217617 (PARI) {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(3-x)^(m-k) + x*O(x^n))), n)}
%o A217617 for(n=0,40,print1(a(n),", "))
%Y A217617 Cf. A217615, A217616, A217461, A216454.
%K A217617 nonn
%O A217617 0,2
%A A217617 _Paul D. Hanna_, Oct 09 2012