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A208237 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + k*x) / (1 + k*x + k^2*x^2).

%I A208237 #20 Nov 02 2014 06:56:00
%S A208237 1,1,2,5,15,54,223,1045,5474,31685,200895,1384470,10304431,82376101,
%T A208237 703949762,6403761365,61784985615,630180031734,6775001385343,
%U A208237 76572619018165,907658144193314,11259399965148005,145879271404693215,1970471655222795990,27702625497930064591
%N A208237 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + k*x) / (1 + k*x + k^2*x^2).
%C A208237 Compare to the identity:
%C A208237 Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + x) / (1 + k*x + k*x^2) = 1/(1-x-x^2).
%C A208237 Compare also to the g.f. of A136127:
%C A208237 x*Sum_{n>=0} n! * x^n * Product_{k=1..n} (2 + k*x) / (1 + 2*k*x + k^2*x^2).
%H A208237 Vaclav Kotesovec, <a href="/A208237/b208237.txt">Table of n, a(n) for n = 0..270</a>
%F A208237 a(n) ~ 2 * 3^(n/2 + 5/4) * n^(n+2) / (exp(n) * Pi^(n+3/2)). - _Vaclav Kotesovec_, Nov 02 2014
%e A208237 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 54*x^5 + 223*x^6 + 1045*x^7 +...
%e A208237 where
%e A208237 A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!*x^2*(1+x)*(1+2*x)/((1+x+x^2)*(1+2*x+4*x^2)) + 3!*x^3*(1+x)*(1+2*x)*(1+3*x)/((1+x+x^2)*(1+2*x+4*x^2)*(1+3*x+9*x^2)) + 4!*x^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/((1+x+x^2)*(1+2*x+4*x^2)*(1+3*x+9*x^2)*(1+4*x+16*x^2)) +...
%o A208237 (PARI) {a(n)=polcoeff( sum(m=0, n, m!*x^m*prod(k=1, m, (1+k*x)/(1+k*x+k^2*x^2 +x*O(x^n))) ), n)}
%o A208237 for(n=0, 30, print1(a(n), ", "))
%Y A208237 Cf. A221370, A204064, A204066, A208236, A136127.
%K A208237 nonn
%O A208237 0,3
%A A208237 _Paul D. Hanna_, Jan 11 2013