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%I #5 Feb 02 2013 23:23:19
%S 1,2,6,24,116,664,4392,32928,276016,2557856,25965408,286538112,
%T 3415359296,43727878528,598510015104,8720853182976,134778021389056,
%U 2202055694727680,37923940767905280,686639853639505920,13038833241899856896,259119925532534413312
%N G.f.: Sum_{n>=0} n! * (2*x)^n * Product_{k=1..n} (1 + k*x)/(1 + 2*k*x + 2*k^2*x^2).
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 116*x^4 + 664*x^5 + 4392*x^6 +...
%e where
%e A(x) = 1 + (2*x)*(1+x)/(1+2*x+2*x^2) + 2!*(2*x)^2*(1+x)*(1+2*x)/((1+2*x+2*x^2)*(1+4*x+8*x^2)) + 3!*(2*x)^3*(1+x)*(1+2*x)*(1+3*x)/((1+2*x+2*x^2)*(1+4*x+8*x^2)*(1+6*x+18*x^2)) + 4!*(2*x)^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/((1+2*x+2*x^2)*(1+4*x+8*x^2)*(1+6*x+18*x^2)*(1+8*x+32*x^2)) +...
%o (PARI) {a(n)=polcoeff( sum(m=0, n, m!*(2*x)^m*prod(k=1, m, (1+k*x)/(1+2*k*x+2*k^2*x^2 +x*O(x^n))) ), n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A221987, A208237, A136127, A221973.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 02 2013
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