A023005 Number of partitions of n into parts of 6 kinds.
1, 6, 27, 98, 315, 918, 2492, 6372, 15525, 36280, 81816, 178794, 380051, 788004, 1597725, 3174210, 6190182, 11867310, 22395359, 41650050, 76413078, 138421358, 247783113, 438616728, 768291650, 1332444330, 2289213495, 3898064226, 6581591157, 11023247880
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. 6th 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*6, 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)^6,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)
Formula
G.f.: Product_{m>=1} 1/(1-x^m)^6.
a(n) ~ exp(2 * Pi * sqrt(n)) / (16 * n^(9/4)). - Vaclav Kotesovec, Feb 28 2015
a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(6*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
Comments