cp's OEIS Frontend

A023004 Number of partitions of n into parts of 5 kinds.

%I A023004 #38 Feb 06 2018 09:18:44
%S A023004 1,5,20,65,190,506,1265,2990,6765,14725,31027,63505,126730,247170,
%T A023004 472295,885723,1633000,2963840,5302075,9358470,16313440,28107365,
%U A023004 47902010,80803485,134992865,223474667,366772720,597049255,964375855,1546208695,2461649861,3892774130
%N A023004 Number of partitions of n into parts of 5 kinds.
%C A023004 a(n) is Euler transform of A010716. - _Alois P. Heinz_, Oct 17 2008
%H A023004 Alois P. Heinz, <a href="/A023004/b023004.txt">Table of n, a(n) for n = 0..1000</a>
%H A023004 Roland Bacher, P. De La Harpe, <a href="https://hal.archives-ouvertes.fr/hal-01285685/document">Conjugacy growth series of some infinitely generated groups</a>. 2016. hal-01285685v2;
%H A023004 P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, A. M. Läuchli, <a href="https://arxiv.org/abs/1601.00958">Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields</a>, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
%H A023004 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A023004 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F A023004 G.f.: Product_{m>=1} 1/(1-x^m)^5.
%F A023004 a(n) ~ 5^(3/2) * exp(Pi * sqrt(10*n/3)) / (32 * 3^(3/2) * n^2) * (1 - (3*sqrt(6/5) /Pi + 5*sqrt(5/6)*Pi/24) / sqrt(n)). - _Vaclav Kotesovec_, Feb 28 2015, extended Jan 16 2017
%F A023004 a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 26 2017
%F A023004 G.f.: exp(5*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%p A023004 with(numtheory): a:=proc(n) option remember; `if`(n=0, 1, add(add(d*5, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # _Alois P. Heinz_, Oct 17 2008
%t A023004 nmax=50; CoefficientList[Series[Product[1/(1-x^k)^5,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Feb 28 2015 *)
%o A023004 (PARI) \ps100 for(n=0,100,print1((polcoeff(1/eta(x)^5,n,x)),","))
%Y A023004 Cf. 5th column of A144064. - _Alois P. Heinz_, Oct 17 2008
%K A023004 nonn
%O A023004 0,2
%A A023004 _David W. Wilson_