A129373 G.f. satisfies: A(x) = (1+x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

1, 1, 1, 2, 3, 4, 7, 9, 13, 19, 26, 34, 52, 67, 89, 123, 166, 214, 295, 380, 501, 660, 858, 1098, 1461, 1858, 2384, 3072, 3940, 4975, 6410, 8070, 10234, 12946, 16322, 20412, 25848, 32201, 40261, 50287, 62728, 77681, 96885, 119673, 148197, 183108, 225974

Offset: 0

Author

Paul D. Hanna, Apr 12 2007

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A074206; A129374, A129375.

Programs

PARI

{a(n)=local(A=1+x);for(i=2,n,A=(1+x)*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); polcoeff(A,n)}

Formula

G.f.: A(x) = Product_{n>=1} (1 + x^n)^A074206(n) where A074206(n) equals the number of ordered factorizations of n.

a(n) ~ exp((1 + 1/r) * (-(1 - 2^(-r)) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-(1 - 2^(-r)) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(2 + 2*r)) / (2^(1/10) * sqrt(Pi) * sqrt(1+r) * n^((2+r)/(2 + 2*r))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018