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A210438 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k^2 + x) / (1 + k^2*x + x^2).

%I A210438 #12 Nov 02 2014 15:36:44
%S A210438 1,1,4,20,174,2262,40894,980590,30095022,1149982990,53521460958,
%T A210438 2980006437662,195562339712590,14936971352026094,1313606920832545022,
%U A210438 131776096083434471678,14956389843996883667182,1906794751364126563388238,271321222225812454677233694
%N A210438 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k^2 + x) / (1 + k^2*x + x^2).
%C A210438 Compare to the identity:
%C A210438 Sum_{n>=0} x^n * Product_{k=1..n} (k + x) / (1 + k*x + x^2) = (1+x^2)/(1-x).
%H A210438 Vaclav Kotesovec, <a href="/A210438/b210438.txt">Table of n, a(n) for n = 0..220</a>
%F A210438 a(n) ~ 2^(2*n+5) * n^(2*n+5/2) / (exp(2*n) * Pi^(2*n+3/2)). - _Vaclav Kotesovec_, Nov 02 2014
%e A210438 G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 174*x^4 + 2262*x^5 + 40894*x^6 +...
%e A210438 where
%e A210438 A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+x)*(4+x)/((1+x+x^2)*(1+4*x+x^2)) + x^3*(1+x)*(4+x)*(9+x)/((1+x+x^2)*(1+4*x+x^2)*(1+9*x+x^2)) + x^4*(1+x)*(4+x)*(9+x)*(16+x)/((1+x+x^2)*(1+4*x+x^2)*(1+9*x+x^2)*(1+16*x+x^2)) +...
%o A210438 (PARI) {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (k^2+x)/(1+k^2*x+x^2 +x*O(x^n))) ), n)}
%o A210438 for(n=0, 30, print1(a(n), ", "))
%Y A210438 Cf. A209778, A221370.
%K A210438 nonn
%O A210438 0,3
%A A210438 _Paul D. Hanna_, Jan 19 2013