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%I #18 Sep 18 2013 10:17:57
%S 1,3,9,30,107,396,1503,5820,22907,91371,368523,1500354,6157669,
%T 25448907,105821865,442409844,1858482563,7840766601,33207750819,
%U 141137445258,601768494201,2573246794374,11032997250357,47421297986868,204286464525165,881900059488741
%N G.f. satisfies: A(x) = (1 + x*(3-x)*A(x)) * (1 + x^2*A(x)).
%C The radius of convergence of the g.f. A(x) is r = 2/(5+sqrt(17)) with A(r) = 2/(1-3*r) = (11+3*sqrt(17))/4.
%C More generally, if A(x) = (1 + x*(t-x)*A(x)) * (1 + x^2*A(x)), |t|>0, then
%C A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k) )
%C where the radius of convergence r of the g.f. A(x) satisfies
%C r = (1-r)^2/(t-r) = (1-t*r)/(2*(1-r)) with A(r) = 1/(r*(1-r)) = 2/(1-t*r).
%H Vincenzo Librandi, <a href="/A216454/b216454.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(3-x)^(n-k) ).
%F G.f.: 2 / ( 1-3*x + sqrt( (1-3*x)^2 - 4*x^3*(3-x) ) ).
%F Recurrence: 3*(n+3)*a(n) = (19*n+30)*a(n-1) - 3*(11*n+3)*a(n-2) + 9*(5*n-6)*a(n-3) - 6*(4*n-9)*a(n-4) + 4*(n-3)*a(n-5). - _Vaclav Kotesovec_, Sep 17 2013
%F a(n) ~ sqrt(7378+1794*sqrt(17)) * ((5+sqrt(17))/2)^n / (16*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Sep 17 2013
%e G.f.: A(x) = 1 + 3*x + 9*x^2 + 30*x^3 + 107*x^4 + 396*x^5 + 1503*x^6 +...
%e The logarithm of the g.f. begins:
%e log(A(x)) = ((3-x) + x)*x + ((3-x)^2 + 2^2*x*(3-x) + x^2)*x^2/2 +
%e ((3-x)^3 + 3^2*x*(3-x)^2 + 3^2*x^2*(3-x) + x^3)*x^3/3 +
%e ((3-x)^4 + 4^2*x*(3-x)^3 + 6^2*x^2*(3-x)^2 + 4^2*x^3*(3-x) + x^4)*x^4/4 +
%e ((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)*x^5/5 +...
%e Explicitly,
%e log(A(x)) = 3*x + 9*x^2/2 + 36*x^3/3 + 149*x^4/4 + 618*x^5/5 + 2592*x^6/6 + 11007*x^7/7 + 47181*x^8/8 + 203634*x^9/9 + 883674*x^10/10 +...
%t CoefficientList[Series[2/(1-3*x+Sqrt[(1-3*x)^2-4*x^3*(3-x)]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 17 2013 *)
%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(3-x)^(m-k) + x*O(x^n)))),n)}
%o (PARI) {a(n)=polcoeff(2/(1-3*x+sqrt((1-3*x)^2-4*x^3*(3-x) +x*O(x^n))),n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A216604, A216434.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 10 2012
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