A023004 Number of partitions of n into parts of 5 kinds.
1, 5, 20, 65, 190, 506, 1265, 2990, 6765, 14725, 31027, 63505, 126730, 247170, 472295, 885723, 1633000, 2963840, 5302075, 9358470, 16313440, 28107365, 47902010, 80803485, 134992865, 223474667, 366772720, 597049255, 964375855, 1546208695, 2461649861, 3892774130
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Roland Bacher, P. De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016. hal-01285685v2;
- P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
- N. J. A. Sloane, Transforms
- Index entries for expansions of Product_{k >= 1} (1-x^k)^m
Crossrefs
Cf. 5th column of A144064. - Alois P. Heinz, Oct 17 2008
Programs
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Maple
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
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Mathematica
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^5,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *) -
PARI
\ps100 for(n=0,100,print1((polcoeff(1/eta(x)^5,n,x)),","))
Formula
G.f.: Product_{m>=1} 1/(1-x^m)^5.
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
a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(5*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
Comments