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%I #7 Jun 15 2014 22:41:17
%S 1,1,8,87,1186,19328,365120,7824589,187217370,4940474068,142398668848,
%T 4447556785374,149541503654196,5382913372109528,206455211385309248,
%U 8402342525589672769,361557591510622222090,16397474363912261372852,781575694749373121466960,39053517651541054156854082
%N G.f.: 1 = Sum_{n>=0} a(n) * x^n*(1-x)^(n+1) / Product_{k=1..n} (1 + 2*(k+2)*x).
%C Triangle T = A243920 is generated by sums of matrix powers of itself such that:
%C T(n,k) = Sum_{j=1..n-k-1} [T^j](n-1,k) with T(n+1,n) = 2*n+1 and T(n,n)=0 for n>=0.
%F a(n) = A243920(n+2,2) / 5.
%e G.f.: 1 = 1*(1-x) + 1*x*(1-x)^2/(1+2*3*x) + 8*x^2*(1-x)^3/((1+2*3*x)*(1+2*4*x)) + 87*x^3*(1-x)^4/((1+2*3*x)*(1+2*4*x)*(1+2*5*x)) + 1186*x^4*(1-x)^5/((1+2*3*x)*(1+2*4*x)*(1+2*5*x)*(1+2*6*x)) +...
%o (PARI) {a(n)=if(n<0, 0, polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x)^(k+1)/prod(j=1, k, 1+2*(j+2)*x+x*O(x^n))), n))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A243920, A243921, A243923, A208677.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 15 2014
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