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

0,3

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

Links

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

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

Example

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 +...

Prog

(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)))}

Crossrefs

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

Cf. A160889, A175926, A330595.

Sequence in context: A267144 A240516 A079386 * A232049 A225658 A226313

Adjacent sequences: A156301 A156302 A156303 * A156305 A156306 A156307

Keyword

nonn

Author

Paul D. Hanna, Feb 08 2009

Status

approved