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%I #71 Oct 09 2023 11:50:44
%S 1,4,14,40,105,252,574,1240,2580,5180,10108,19208,35693,64960,116090,
%T 203984,353017,602348,1014580,1688400,2778517,4524760,7296752,
%U 11658920,18468245,29015700,45235414,70005376,107585845,164245380,249162620,375704920,563251038
%N Number of partitions of n into parts of 4 kinds.
%C a(n) is Euler transform of A010709. - _Alois P. Heinz_, Oct 17 2008
%H Seiichi Manyama, <a href="/A023003/b023003.txt">Table of n, a(n) for n = 0..10000</a> (first 501 terms from T. D. Noe)
%H Roland Bacher and 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 Vaclav Kotesovec, <a href="https://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
%H P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, and 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)^4.
%F a(0)=1, a(n) = (1/n) * Sum_{k=0..n-1} 4*a(k)*sigma_1(n-k). - _Joerg Arndt_, Feb 05 2011
%F a(n) ~ exp(2 * Pi * sqrt(2*n/3)) / (2^(7/4) * 3^(5/4) * n^(7/4)) * (1 - (35*sqrt(3)/(16*Pi) + Pi/(3*sqrt(3))) / sqrt(n)). - _Vaclav Kotesovec_, Feb 28 2015, extended Jan 16 2017
%F G.f.: exp(4*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%F From _Peter Bala_, Oct 05 2023: (Start)
%F The even bisection of the g.f. A(x) is (A(x) + A(-x))/2 = 1 + 14*x^2 + 105*x^4 + 574*x^6 + ... = Product_{n >= 1} (1 + x^(2*n))^14 / (1 - x^(8*n))^4 = F(x^2)*A(x^8), where F(x) = Product_{n >= 1} (1 + x^n)^14 is the g.f. of A022579.
%F The odd bisection of the g.f. is (A(x) - A(-x))/2 = 4*x + 40*x^3 + 252*x^5 + 1240*x^7 + ... = (4*x) * Product_{n >= 1} (1 + x^(2*n))^2 * (1 - x^(8*n))^4 / (1 - x^(2*n))^8. (End)
%p with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*4, 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)^4,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Feb 28 2015 *)
%t CoefficientList[1/QPochhammer[x]^4 + O[x]^40, x] (* _Jean-François Alcover_, Jan 31 2016 *)
%o (PARI) \ps100
%o for(n=0,100,print1((polcoeff(1/eta(x)^4,n,x)),","))
%o (Julia) # DedekindEta is defined in A000594.
%o A023003List(len) = DedekindEta(len, -4)
%o A023003List(33) |> println # _Peter Luschny_, Mar 10 2018
%Y 4th column of A144064.
%K nonn,easy
%O 0,2
%A _David W. Wilson_
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