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A023005 Number of partitions of n into parts of 6 kinds. 4

%I

%S 1,6,27,98,315,918,2492,6372,15525,36280,81816,178794,380051,788004,

%T 1597725,3174210,6190182,11867310,22395359,41650050,76413078,

%U 138421358,247783113,438616728,768291650,1332444330,2289213495,3898064226,6581591157,11023247880

%N Number of partitions of n into parts of 6 kinds.

%C a(n) is Euler transform of A010722. - _Alois P. Heinz_, Oct 17 2008

%H Alois P. Heinz, <a href="/A023005/b023005.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>

%F G.f.: Product_{m>=1} 1/(1-x^m)^6.

%F a(n) ~ exp(2 * Pi * sqrt(n)) / (16 * n^(9/4)). - _Vaclav Kotesovec_, Feb 28 2015

%F a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 26 2017

%F G.f.: exp(6*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018

%p 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

%t nmax=50; CoefficientList[Series[Product[1/(1-x^k)^6,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Feb 28 2015 *)

%Y Cf. 6th column of A144064. - _Alois P. Heinz_, Oct 17 2008

%K nonn

%O 0,2

%A _David W. Wilson_

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Last modified July 25 16:39 EDT 2021. Contains 346291 sequences. (Running on oeis4.)