A156304 G.f.: A(x) = exp( Sum_{n>=1} sigma(n^3)*x^n/n ), a power series in x with integer coefficients.

1, 1, 8, 21, 77, 199, 661, 1663, 4852, 12382, 33289, 82877, 213026, 518109, 1279852, 3053404, 7312985, 17093793, 39952528, 91661695, 209709116, 473095589, 1062567288, 2359804486, 5214774263, 11415904502, 24860918943, 53709881911

Offset: 0

Author

Paul D. Hanna, Feb 08 2009

Keywords

• nonn

Comments

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Euler transform of A160889. - Vaclav Kotesovec, Nov 01 2024

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 21*x^3 + 77*x^4 + 199*x^5 + 661*x^6 +...
log(A(x)) = x + 15*x^2/2 + 40*x^3/3 + 127*x^4/4 + 156*x^5/5 + 600*x^6/6 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A156303, A202993, A203557.

Cf. A160889, A175926, A330595.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1,n,sigma(m^3)*x^m/m)+x*O(x^n)),n)}

PARI

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k^3)*a(n-k)))}

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k^3) * a(n-k) for n>0, with a(0)=1.

log(a(n)) ~ 4*Pi*c^(1/4)*n^(3/4) / (3^(5/4)*5^(1/4)), where c = A330595 = Product_{primes p} (1 + 1/p^2 + 1/p^3) = 1.74893299784324530303390699... - Vaclav Kotesovec, Nov 01 2024