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%I #35 Sep 08 2022 08:44:48
%S 1,1,3,12,82,725,8811,128340,2257687,45658174,1052672116,27108596725,
%T 772945749970,24137251258926,819742344728692,30069017799172228,
%U 1184889562926838573,49914141857616862435
%N Number of partitions in expanding space: sigma(n,q) is the sum of the q-th powers of the divisors of n.
%H Vaclav Kotesovec, <a href="/A023881/b023881.txt">Table of n, a(n) for n = 0..370</a>
%F G.f.: exp( Sum_{k>0} sigma_k(k) * x^k / k). - _Michael Somos_, Feb 15 2006
%F G.f.: Product_{n>=1} (1 - n^n*x^n)^(-1/n). - _Paul D. Hanna_, Mar 08 2011
%F a(n) ~ n^(n-1). - _Vaclav Kotesovec_, Oct 08 2016
%e 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 + ...
%p 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
%t 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 *)
%o (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 */
%o (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_ */
%o (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
%Y Cf. A023882, A023887, A158952.
%K nonn
%O 0,3
%A _Olivier Gérard_
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