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

Author

Olivier Gérard

Keywords

• nonn

Examples

			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 + ...
		

Links

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

Crossrefs

Cf. A023882, A023887, A158952.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-k^k*x^k)^(1/k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

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 *)

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 */

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