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A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + k*x) / (1 + n*x + k*x^2).

%I A204066 #11 Nov 02 2014 06:17:12
%S A204066 1,1,4,16,82,502,3574,29002,264166,2668666,29612014,358025986,
%T A204066 4684916902,65966957722,994546450174,15984888286642,272845934899606,
%U A204066 4929166716321706,93963635086523374,1884915966747571906,39691711412770983622,875410001054417122042,20180907494704416823774
%N A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + k*x) / (1 + n*x + k*x^2).
%H A204066 Vaclav Kotesovec, <a href="/A204066/b204066.txt">Table of n, a(n) for n = 0..250</a>
%F A204066 a(n) ~ exp(1/2) * n! * n/2. - _Vaclav Kotesovec_, Nov 02 2014
%e A204066 G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 82*x^4 + 502*x^5 + 3574*x^6 +...
%e A204066 where
%e A204066 A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(2+x)*(2+2*x)/((1+2*x+x^2)*(1+2*x+2*x^2)) + x^3*(3+x)*(3+2*x)*(3+3*x)/((1+3*x+x^2)*(1+3*x+2*x^2)*(1+3*x+3*x^2)) + x^4*(4+x)*(4+2*x)*(4+3*x)*(4+4*x)/((1+4*x+x^2)*(1+4*x+2*x^2)*(1+4*x+3*x^2)*(1+4*x+4*x^2)) +...
%o A204066 (PARI) {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (m+k*x)/(1+m*x+k*x^2 +x*O(x^n))) ), n)}
%o A204066 for(n=0, 30, print1(a(n), ", "))
%Y A204066 Cf. A204064, A187741, A187742, A220181.
%K A204066 nonn
%O A204066 0,3
%A A204066 _Paul D. Hanna_, Jan 09 2013