OFFSET
0,3
COMMENTS
Compare to the g.f. of A000081: G(x) = exp( Sum_{n>=1} G(x^n)*x^n/n ).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..81
FORMULA
From Seiichi Manyama, Jun 01 2023: (Start)
A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(k+1))^(2^k * a(k)).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * 2^(d-1) * a(d-1) ) * a(n-k). (End)
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 138*x^4 + 2370*x^5 +...
log(A(x)) = A(2x)*x + A(2x^2)*x^2/2 + A(2x^3)*x^3/3 + A(2x^4)*x^4/4 + A(2x^5)*x^5/5 +...
More generally, if F(x,q) = exp( Sum_{n>=1} F(q*x^n,q)*x^n/n )
then coefficients in F(x,q) = Sum_{n>=0} c(n,q)*x^n begin:
c(0,q) = 1; c(1,q) = 1; c(2,q) = q + 1;
c(3,q) = q^3 + q^2 + q + 1;
c(4,q) = q^6 + q^5 + q^4 + 2*q^3 + 3/2*q^2 + 3/2*q + 1;
c(5,q) = q^10 + q^9 + q^8 + 2*q^7 + 5/2*q^6 + 5/2*q^5 + 3*q^4 + 3*q^3 + 3/2*q^2 + 3/2*q + 1; ...
where C(n,q) are integers for integer values of q.
PROG
(PARI) {a(n)=my(A=1+x); for(i=1, n, A=exp(sum(m=1, n, subst(A, x, 2*x^m+x*O(x^n))*x^m/m))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 15 2010
STATUS
approved