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G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^k)^n) ).
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%I #20 Apr 09 2016 02:23:53

%S 1,1,2,4,7,14,25,44,79,137,237,408,689,1162,1946,3231,5342,8776,14340,

%T 23326,37758,60847,97670,156145,248697,394719,624343,984360,1547187,

%U 2424581,3788730,5904230,9176723,14226914,22002523,33947526,52258177,80268131,123028407

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

%C Compare to the dual g.f. of A219229:

%C exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^n)^k) ).

%H Paul D. Hanna and Vaclav Kotesovec, <a href="/A218576/b218576.txt">Table of n, a(n) for n = 0..2000</a> (terms 0..1000 from Paul D. Hanna)

%F Conjecture: a(n) ~ c * d^n, where d = A060006 = 1.3247179572447... is the real root of the equation d*(d^2-1) = 1 and c = 43328430766.390... . - _Vaclav Kotesovec_, Apr 09 2016

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 25*x^6 + 44*x^7 +...

%e where

%e log(A(x)) = x/1*((1+x*(1+x))*(1+x^2*(1+x^2))*(1+x^3*(1+x^3))*...) +

%e x^2/2*((1+x^2*(1+x)^2)*(1+x^4*(1+x^2)^2)*(1+x^6*(1+x^3)^2)*...) +

%e x^3/3*((1+x^3*(1+x)^3)*(1+x^6*(1+x^2)^3)*(1+x^9*(1+x^3)^3)*...) +

%e x^4/4*((1+x^4*(1+x)^4)*(1+x^8*(1+x^2)^4)*(1+x^12*(1+x^3)^4)*...) +...

%e Explicitly,

%e log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 11*x^4/4 + 26*x^5/5 + 39*x^6/6 + 57*x^7/7 + 99*x^8/8 + 142*x^9/9 + 208*x^10/10 +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*prod(k=1,n\m,(1+x^(m*k)*(1+x^k+x*O(x^n))^m )))),n)}

%o for(n=0,50,print1(a(n),", "))

%Y Cf. A219229, A218575, A218552, A218153.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 02 2012