A023881 Number of partitions in expanding space: sigma(n,q) is the sum of the q-th powers of the divisors of n.

1, 1, 3, 12, 82, 725, 8811, 128340, 2257687, 45658174, 1052672116, 27108596725, 772945749970, 24137251258926, 819742344728692, 30069017799172228, 1184889562926838573, 49914141857616862435

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..370

Formula

G.f.: exp( Sum_{k>0} sigma_k(k) * x^k / k). - Michael Somos, Feb 15 2006

G.f.: Product_{n>=1} (1 - n^n*x^n)^(-1/n). - Paul D. Hanna, Mar 08 2011

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

Example

G.f. = 1 + x + 3*x^2 + 12*x^3 + 82*x^4 + 725*x^5 + 8811*x^6 + 128340*x^7 + 2257687*x^8 + ...

Maple

seq(coeff(series(mul((1-k^k*x^k)^(-1/k), k=1..n), x, n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

Mathematica

nmax=30; CoefficientList[Series[Product[1/(1-k^k*x^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Oct 31 2018 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( exp( sum( k=1, n, sigma(k, k) * x^k / k, x * O(x^n))), n))} /* Michael Somos, Feb 15 2006 */
(PARI) {a(n)=if(n<0, 0, polcoeff(prod(k=1, n, (1-k^k*x^k+x*O(x^n))^(-1/k)), n))} /* Paul D. Hanna */
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-k^k*x^k)^(1/k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

Crossrefs

Cf. A023882, A023887, A158952.

Sequence in context: A188227 A357665 A224608 * A067111 A171186 A229421

Adjacent sequences: A023878 A023879 A023880 * A023882 A023883 A023884

Keyword

nonn

Author

Olivier Gérard

Status

approved