A023003 Number of partitions of n into parts of 4 kinds.
1, 4, 14, 40, 105, 252, 574, 1240, 2580, 5180, 10108, 19208, 35693, 64960, 116090, 203984, 353017, 602348, 1014580, 1688400, 2778517, 4524760, 7296752, 11658920, 18468245, 29015700, 45235414, 70005376, 107585845, 164245380, 249162620, 375704920, 563251038
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from T. D. Noe)
- Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, 2016, hal-01285685v2.
- Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
- P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, and 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
4th column of A144064.
Programs
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Julia
# DedekindEta is defined in A000594. A023003List(len) = DedekindEta(len, -4) A023003List(33) |> println # Peter Luschny, Mar 10 2018
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Maple
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
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Mathematica
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^4,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *) CoefficientList[1/QPochhammer[x]^4 + O[x]^40, x] (* Jean-François Alcover, Jan 31 2016 *) -
PARI
\ps100 for(n=0,100,print1((polcoeff(1/eta(x)^4,n,x)),","))
Formula
G.f.: Product_{m>=1} 1/(1-x^m)^4.
a(0)=1, a(n) = (1/n) * Sum_{k=0..n-1} 4*a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011
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
G.f.: exp(4*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
From Peter Bala, Oct 05 2023: (Start)
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.
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)
Comments