A023007 Number of partitions of n into parts of 8 kinds.
1, 8, 44, 192, 726, 2464, 7704, 22528, 62337, 164560, 417140, 1020416, 2418710, 5573568, 12520744, 27484160, 59068372, 124505880, 257770964, 524871424, 1052316364, 2079491744, 4053978040, 7803219968, 14840711765, 27907041392, 51917588800, 95608651776
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
- 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. 8th 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*8, 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)^8,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)
Formula
a(n) ~ exp(4 * Pi * sqrt(n/3)) / (sqrt(2) * 3^(9/4) * n^(11/4)). - Vaclav Kotesovec, Feb 28 2015
a(0) = 1, a(n) = (8/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
Comments