A219260 - OEIS

%I #10 Nov 17 2012 10:02:54

%S 1,1,3,10,37,143,576,2393,10178,44133,194341,866867,3908454,17784385,

%T 81562890,376637216,1749704080,8171760933,38346326963,180707395127,

%U 854850922373,4057990958069,19324260613400,92288612451684,441919933724974,2121281845071105,10205443975074195

%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^k)^n)).

%C Compare to the dual g.f. G(x) of A219261:

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

%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 143*x^5 + 576*x^6 + 2393*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^3)^2)*...) +

%e x^3*A(x)^3/3*((1+x^3*A(x)^3)*(1+x^6*A(x^2)^3)*(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^2)^4)*(1+x^12*A(x^3)^4)*...) +...

%e Explicitly,

%e log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 101*x^4/4 + 461*x^5/5 + 2144*x^6/6 + 10109*x^7/7 + 48117*x^8/8 + 230998*x^9/9 + 1115875*x^10/10 +...

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

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

%Y Cf. A218552, A219262, A219261, A218153.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 16 2012