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%I #6 Mar 30 2012 18:37:37
%S 1,1,4,23,169,1496,15400,180055,2350867,33840345,531707256,9045486916,
%T 165507986668,3238945135696,67470601883224,1489923969768999,
%U 34753006977085479,853544188578784147,22011310309759024484,594514290559650994575,16780116115165946427561
%N G.f.: 1 = Sum_{n>=1} a(n) * x^n*(1-x)^(n+1) / Product_{k=1..n} (1 + (k+1)*x).
%C Related triangle T=A132623 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) = n+1 and T(n,n)=0 for n>=0.
%F a(n) = A132623(n+1, 1) / 2.
%e 1 = 1*(1-x) + 1*x*(1-x)^2/(1+2*x) + 4*x^2*(1-x)^3/((1+2*x)*(1+3*x)) + 23*x^3*(1-x)^4/((1+2*x)*(1+3*x)*(1+4*x)) + 169*x^4*(1-x)^5/((1+2*x)*(1+3*x)*(1+4*x)*(1+5*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+(j+1)*x+x*O(x^n))), n))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A132623, A132624, A208677, A208678.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 14 2012
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