A219231 - OEIS

%I #12 Nov 16 2012 00:40:19

%S 1,1,2,5,15,47,160,554,1987,7243,26873,100930,383412,1469673,5679033,

%T 22095308,86489211,340360513,1345814572,5344184197,21303295069,

%U 85216434084,341960332173,1376212103798,5553269024152,22463340663474,91071265881382,369996643180885,1506118767637576

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

%C Compare to the dual g.f. G(x) of A218551:

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

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 47*x^5 + 160*x^6 + 554*x^7 +...

%e where

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

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

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

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

%e Explicitly,

%e log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 39*x^4/4 + 146*x^5/5 + 594*x^6/6 + 2346*x^7/7 + 9543*x^8/8 + 38710*x^9/9 + 158448*x^10/10 +...

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m/m*prod(k=1, n\m+1, 1/(1-x^(m*k)*subst(A, x, x^m +x*O(x^n))^k))))); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A218551, A219230, A219229, A219232, A218153.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 16 2012