A205814 G.f.: Product_{n>=1} [ (1 - 2^n*x^n) / (1 - (n+2)^n*x^n) ]^(1/n).

1, 1, 9, 54, 482, 4239, 55561, 785554, 14133055, 285547760, 6666380256, 172748192767, 4974178683908, 156462697434990, 5354832107694444, 197710292330150160, 7839473395324929677, 332071887435037103895, 14968498613432649146050, 715294449027151380463781

Offset: 0

Author

Paul D. Hanna, Feb 01 2012

Keywords

• nonn

Comments

Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.

Examples

			G.f.: A(x) = 1 + x + 9*x^2 + 54*x^3 + 482*x^4 + 4239*x^5 + 55561*x^6 +...
where the g.f. equals the product:
A(x) = (1-2*x)/(1-3*x) * ((1-2^2*x^2)/(1-4^2*x^2))^(1/2) * ((1-2^3*x^3)/(1-5^3*x^3))^(1/3) * ((1-2^4*x^4)/(1-6^4*x^4))^(1/4) * ((1-2^5*x^5)/(1-7^5*x^5))^(1/5) *...
The logarithm equals the l.g.f. of A205815:
log(A(x)) = x + 17*x^2/2 + 136*x^3/3 + 1585*x^4/4 + 16986*x^5/5 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Crossrefs

Cf. A205815 (log), A205811, A023881.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=1, m, binomial(m, k)*sigma(m, k)*2^(m-k))+x*O(x^n))), n)}

PARI

{a(n)=polcoeff(prod(k=1, n, ((1-2^k*x^k)/(1-(k+2)^k*x^k +x*O(x^n)))^(1/k)), n)}

Formula

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 2^(n-k) ).

a(n) ~ exp(2) * n^(n-1). - Vaclav Kotesovec, Oct 08 2016