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A206765 G.f.: Product_{n>=1} [ (1 - 3^n*x^n) / (1 - (n+3)^n*x^n) ]^(1/n).

%I A206765 #14 Feb 12 2018 21:06:08
%S A206765 1,1,12,87,907,8393,118932,1683990,31209334,635005549,15054451057,
%T A206765 393600573337,11466736952722,363842430190308,12564913404375244,
%U A206765 467483278911401155,18670853023655302285,795978439482823960066,36093307429580735618893
%N A206765 G.f.: Product_{n>=1} [ (1 - 3^n*x^n) / (1 - (n+3)^n*x^n) ]^(1/n).
%C A206765 Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.
%H A206765 Vaclav Kotesovec, <a href="/A206765/b206765.txt">Table of n, a(n) for n = 0..380</a>
%F A206765 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 3^(n-k) ).
%F A206765 Logarithmic derivative yields A206766.
%F A206765 a(n) ~ exp(3) * n^(n-1). - _Vaclav Kotesovec_, Oct 08 2016
%e A206765 G.f.: A(x) = 1 + x + 12*x^2 + 87*x^3 + 907*x^4 + 8393*x^5 + 118932*x^6 +...
%e A206765 where the g.f. equals the product:
%e A206765 A(x) = (1-3*x)/(1-4*x) * ((1-3^2*x^2)/(1-5^2*x^2))^(1/2) * ((1-3^3*x^3)/(1-6^3*x^3))^(1/3) * ((1-3^4*x^4)/(1-7^4*x^4))^(1/4) * ((1-3^5*x^5)/(1-8^5*x^5))^(1/5) *...
%e A206765 The logarithm equals the l.g.f. of A206766:
%e A206765 log(A(x)) = x + 23*x^2/2 + 226*x^3/3 + 3039*x^4/4 + 33306*x^5/5 +...
%t A206765 max = 19; p = Product[((1-3^n*x^n) / (1-(n+3)^n*x^n))^(1/n), {n, 1, max}] + O[x]^max; CoefficientList[p, x] (* _Jean-François Alcover_, Oct 08 2016 *)
%o A206765 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=1, m, binomial(m, k)*sigma(m, k)*3^(m-k))+x*O(x^n))), n)}
%o A206765 (PARI) {a(n)=polcoeff(prod(k=1, n, ((1-3^k*x^k)/(1-(k+3)^k*x^k +x*O(x^n)))^(1/k)), n)}
%o A206765 for(n=0,31,print1(a(n),", "))
%Y A206765 Cf. A206766 (log), A205814, A205811, A206763.
%K A206765 nonn
%O A206765 0,3
%A A206765 _Paul D. Hanna_, Feb 12 2012