OFFSET
0,3
FORMULA
G.f. satisfies: A(x) = exp( Sum_{n>=1} (x^n/n)*Sum_{d|n} d*A(x)^(n*d) ).
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 59*x^4 + 328*x^5 + 1987*x^6 +...
The g.f. A = A(x) satisfies the product:
A = 1/((1 - x*A)*(1 - x^2*A^4)*(1 - x^3*A^9)*(1 - x^4*A^16)*...)
as well as the logarithmic series:
log(A) = x*A + x^2*(A^2 + 2*A^4)/2 + x^3*(A^3 + 3*A^9)/3 + x^4*(A^4 + 2*A^8 + 4*A^16)/4 + x^5*(A^5 + 5*A^25)/5 + x^6*(A^6 + 2*A^12 + 3*A^18 + 6*A^36)/6 +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, 1-x^k*A^(k^2)+x*O(x^n))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (x^m/m)*sumdiv(m, d, d*A^(m*d))+x*O(x^n)))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 09 2011
STATUS
approved