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

0,2

Comments

a(n) is Euler transform of A010709. - Alois P. Heinz, Oct 17 2008

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

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)

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

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 *)

Prog

(PARI) \ps100
for(n=0, 100, print1((polcoeff(1/eta(x)^4, n, x)), ", "))
(Julia) # DedekindEta is defined in A000594.
A023003List(len) = DedekindEta(len, -4)
A023003List(33) |> println # Peter Luschny, Mar 10 2018

Crossrefs

4th column of A144064.

Sequence in context: A278680 A121593 A160527 * A001872 A054443 A281766

Adjacent sequences: A023000 A023001 A023002 * A023004 A023005 A023006

Keyword

nonn,easy

Author

David W. Wilson

Status

approved