OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..3337
FORMULA
G.f.: A(x) = Product_{n>=1} (1/x)*Series_Reversion( x/(1 + x^n) ); equivalently, G.f.: A(x) = Product_{n>=1} G(x^n,n) where G(x,n) = 1 + x*G(x,n)^n.
a(n) ~ c * 2^n / n^(3/2), where c = 2.8176325363130737043447... if n is even and c = 1.784372019603712867208... if n is odd. - Vaclav Kotesovec, Jan 15 2019
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 18*x^6 +...
where A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), or
A(x) = exp(x + 3/2*x^2 + 4/3*x^3 + 11/4*x^4 + 6/5*x^5 +...).
The g.f. can also be expressed as the product:
A(x) = 1/(1-x)*G000108(x^2)*G001764(x^3)*G002293(x^4)*G002294(x^5)*...
where the functions are g.f.s of well-known sequences:
G000108(x) = 1 + x*G000108(x)^2 = g.f. of A000108 ;
G001764(x) = 1 + x*G001764(x)^3 = g.f. of A001764 ;
G002293(x) = 1 + x*G002293(x)^4 = g.f. of A002293 ;
G002294(x) = 1 + x*G002294(x)^5 = g.f. of A002294 ; etc.
PROG
(PARI) {a(n)=polcoeff(exp(x*Ser(vector(n, m, sumdiv(m, d, binomial(m, d))/m))+x*O(x^n)), n)}
(PARI) {a(n)=polcoeff(prod(m=1, n, 1/x*serreverse(x/(1+x^m +x*O(x^n)))), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 20 2005, Nov 10 2007
STATUS
approved