A218552 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^k)^n) ).

1, 1, 2, 4, 9, 20, 46, 107, 253, 604, 1463, 3573, 8812, 21901, 54837, 138145, 350068, 891529, 2281092, 5860471, 15113614, 39109461, 101521521, 264286160, 689820642, 1804890193, 4733051924, 12437565725, 32746931264, 86375236835, 228212881032, 603915863737, 1600500761487

Offset: 0

Author

Paul D. Hanna, Nov 01 2012

Keywords

• nonn

Comments

Compare to the dual g.f. G(x) of A219232:

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

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 107*x^7 +...
where
log(A(x)) = x/1*((1+x*A(x))*(1+x^2*A(x^2))*(1+x^3*A(x^3))*...) +
x^2/2*((1+x^2*A(x)^2)*(1+x^4*A(x^2)^2)*(1+x^6*A(x^3)^2)*...) +
x^3/3*((1+x^3*A(x)^3)*(1+x^6*A(x^2)^3)*(1+x^9*A(x^3)^3)*...) +
x^4/4*((1+x^4*A(x)^4)*(1+x^8*A(x^2)^4)*(1+x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 117*x^6/6 + 295*x^7/7 + 755*x^8/8 + 1933*x^9/9 + 5048*x^10/10 +...
		

Crossrefs

Cf. A219232, A218551, A218153, A218576.

Programs

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+x^(m*k)*subst(A,x,x^k +x*O(x^n))^m))));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))