cp's OEIS Frontend

A023020 Number of partitions of n into parts of 22 kinds.

%I A023020 #34 Jun 28 2025 08:46:46
%S A023020 1,22,275,2530,18975,122430,702328,3661900,17627775,79264900,
%T A023020 335937954,1351507830,5191041625,19125838600,67862904725,232671319474,
%U A023020 773027485065,2494957906100,7839428942950,24025993453000,71941861591215
%N A023020 Number of partitions of n into parts of 22 kinds.
%C A023020 a(n) is Euler transform of A010861. - _Alois P. Heinz_, Oct 17 2008
%H A023020 Seiichi Manyama, <a href="/A023020/b023020.txt">Table of n, a(n) for n = 0..1000</a>
%H A023020 M. V. Movshev, <a href="https://arxiv.org/abs/1602.04673">A formula for the partition function of the beta-gamma system on the cone pure spinors</a>, arXiv preprint arXiv:1602.04673 [hep-th], 2016. [Gives sequence that appears to agree with this one]
%H A023020 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a> [_Alois P. Heinz_, Oct 17 2008]
%H A023020 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F A023020 G.f.: Product_{m>=1} 1/(1-x^m)^22.
%F A023020 a(0) = 1, a(n) = (22/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 27 2017
%F A023020 a(n) ~ m^((m+1)/4) * exp(Pi*sqrt(2*m*n/3)) / (2^((3*m+5)/4) * 3^((m+1)/4) * n^((m+3)/4)) * (1 - ((9+Pi^2)*m^2+36*m+27) / (24*Pi*sqrt(6*m*n))), set m = 22. - _Vaclav Kotesovec_, Jun 28 2025
%p A023020 with (numtheory): a:= proc(n) option remember; `if`(n=0, 1, add (add (d*22, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq (a(n), n=0..40); # _Alois P. Heinz_, Oct 17 2008
%t A023020 CoefficientList[Series[1/QPochhammer[x]^22, {x, 0, 30}], x] (* _Indranil Ghosh_, Mar 27 2017 *)
%o A023020 (PARI) Vec(1/eta(x)^22 + O(x^30)) \\ _Indranil Ghosh_, Mar 27 2017
%Y A023020 Cf. 22nd column of A144064. - _Alois P. Heinz_, Oct 17 2008
%K A023020 nonn
%O A023020 0,2
%A A023020 _David W. Wilson_