A219261 - OEIS

%I #7 Nov 16 2012 19:34:32

%S 1,1,3,9,33,124,503,2089,8960,39142,173978,783347,3567123,16395199,

%T 75966835,354447193,1663921966,7853325055,37244059607,177388171005,

%U 848148206917,4069483589180,19588001935380,94559416543623,457697632011720,2220852281129195,10800560004895426

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

%C Compare to the dual g.f. G(x) of A219260:

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

%e G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 33*x^4 + 124*x^5 + 503*x^6 + 2089*x^7 +...

%e where

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

%e x^2*A(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*A(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*A(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 + 5*x^2/2 + 19*x^3/3 + 89*x^4/4 + 396*x^5/5 + 1895*x^6/6 + 8989*x^7/7 + 43545*x^8/8 + 211645*x^9/9 + 1036560*x^10/10 +...

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

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

%Y Cf. A219232, A219263, A219260, A218153.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 16 2012