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A230682 O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k^2 + x)/(1 + k^2*x).

%I A230682 #10 Oct 28 2014 16:56:48
%S A230682 1,1,4,21,181,2320,41581,991821,30339364,1156828681,53761779721,
%T A230682 2990342767680,196097039232121,14969727522159481,1315952342285654884,
%U A230682 131970189920614495581,14974773731779775857021,1908770813250950767227280,271560466483540753565395621
%N A230682 O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k^2 + x)/(1 + k^2*x).
%C A230682 Compare to an o.g.f. of Genocchi numbers of the first kind (A110501):
%C A230682 Sum_{n>=0} x^n * Product_{k=1..n} k^2/(1 + k^2*x).
%C A230682 Also, compare to a g.f. of Fibonacci numbers (A000045):
%C A230682 Sum_{n>=0} x^n * Product_{k=1..n} (k + x)/(1 + k*x).
%H A230682 Vaclav Kotesovec, <a href="/A230682/b230682.txt">Table of n, a(n) for n = 0..230</a>
%F A230682 a(n) ~ 2^(2*n+5) * n^(2*n+5/2) / (Pi^(2*n+3/2) * exp(2*n)). - _Vaclav Kotesovec_, Oct 28 2014
%e A230682 G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 181*x^4 + 2320*x^5 + 41581*x^6 +...
%e A230682 where
%e A230682 A(x) = 1 + x*(1+x)/(1+x) + x^2*(1+x)*(4+x)/((1+x)*(1+4*x)) + x^3*(1+x)*(4+x)*(9+x)/((1+x)*(1+4*x)*(1+9*x)) + x^4*(1+x)*(4+x)*(9+x)*(16+x)/((1+x)*(1+4*x)*(1+9*x)*(1+16*x)) +...
%o A230682 (PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, k^2+x+x*O(x^n))/prod(k=1, m, 1+k^2*x+x*O(x^n))), n)}
%o A230682 for(n=0, 20, print1(a(n), ", "))
%Y A230682 Cf. A230740.
%K A230682 nonn
%O A230682 0,3
%A A230682 _Paul D. Hanna_, Oct 27 2013