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A219229 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^n)^k) ).

%I A219229 #12 Nov 16 2012 00:42:33
%S A219229 1,1,2,4,9,20,45,97,217,476,1043,2256,4890,10540,22695,48614,103856,
%T A219229 221103,469738,995299,2104514,4439410,9345471,19631897,41162937,
%U A219229 86148416,179987656,375411910,781773332,1625473202,3374684346,6996168098,14483902403,29945326755
%N A219229 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^n)^k) ).
%C A219229 Compare to the dual g.f. of A218576:
%C A219229 exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^k)^n) ).
%H A219229 Paul D. Hanna, <a href="/A219229/b219229.txt">Table of n, a(n) for n = 0..1000</a>
%e A219229 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 45*x^6 + 97*x^7 +...
%e A219229 where
%e A219229 log(A(x)) = x/1*((1+x*(1+x))*(1+x^2*(1+x)^2)*(1+x^3*(1+x)^3)*...) +
%e A219229 x^2/2*((1+x^2*(1+x^2))*(1+x^4*(1+x^2)^2)*(1+x^6*(1+x^2)^3)*...) +
%e A219229 x^3/3*((1+x^3*(1+x^3))*(1+x^6*(1+x^3)^2)*(1+x^9*(1+x^3)^3)*...) +
%e A219229 x^4/4*((1+x^4*(1+x^4))*(1+x^8*(1+x^4)^2)*(1+x^12*(1+x^4)^3)*...) +...
%e A219229 Explicitly,
%e A219229 log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 111*x^6/6 + 232*x^7/7 + 555*x^8/8 + 1204*x^9/9 + 2608*x^10/10 +...
%o A219229 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*prod(k=1, n\m, (1+x^(m*k)*(1+x^m+x*O(x^n))^k )))), n)}
%o A219229 for(n=0, 50, print1(a(n), ", "))
%Y A219229 Cf. A218576, A219230, A219231.
%K A219229 nonn
%O A219229 0,3
%A A219229 _Paul D. Hanna_, Nov 15 2012