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A218575 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*(1 + x^k)^n) ).

%I A218575 #13 Nov 15 2012 23:47:34
%S A218575 1,1,2,5,11,26,56,125,269,578,1228,2600,5447,11366,23575,48664,99950,
%T A218575 204383,416196,844299,1706368,3436555,6898255,13803732,27539833,
%U A218575 54788703,108703105,215112006,424628345,836218453,1643005834,3221104945,6301628342,12303151494
%N A218575 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*(1 + x^k)^n) ).
%C A218575  Compare to the dual g.f. of A219230:
%C A218575 exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*(1 + x^n)^k) ).
%H A218575 Paul D. Hanna, <a href="/A218575/b218575.txt">Table of n, a(n) for n = 0..1000</a>
%e A218575 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 26*x^5 + 56*x^6 + 125*x^7 +...
%e A218575 where
%e A218575 log(A(x)) = x/(1*(1-x*(1+x))*(1-x^2*(1+x^2))*(1-x^3*(1+x^3))*...) +
%e A218575 x^2/(2*(1-x^2*(1+x)^2)*(1-x^4*(1+x^2)^2)*(1-x^6*(1+x^3)^2)*...) +
%e A218575 x^3/(3*(1-x^3*(1+x)^3)*(1-x^6*(1+x^2)^3)*(1-x^9*(1+x^3)^3)*...) +
%e A218575 x^4/(4*(1-x^4*(1+x)^4)*(1-x^8*(1+x^2)^4)*(1-x^12*(1+x^3)^4)*...) +...
%e A218575 Explicitly,
%e A218575 log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 23*x^4/4 + 61*x^5/5 + 120*x^6/6 + 274*x^7/7 + 527*x^8/8 + 1054*x^9/9 + 1973*x^10/10 + 3807*x^11/11 + 6824*x^12/12 +...
%o A218575 (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*prod(k=1,n\m,1/(1-x^(m*k)*(1+x^k)^m +x*O(x^n))))),n)}
%o A218575 for(n=0,40,print1(a(n),", "))
%Y A218575 Cf. A218576, A218551, A219230.
%K A218575 nonn
%O A218575 0,3
%A A218575 _Paul D. Hanna_, Nov 02 2012