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A023003 Number of partitions of n into parts of 4 kinds. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

Roland Bacher, P De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016. hal-01285685v2; https://hal.archives-ouvertes.fr/hal-01285685/document

P Nataf, M Lajkó, A Wietek, K Penc, F Mila, AM Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958, 2016

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from T. D. Noe)

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.

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

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)), ", "))

CROSSREFS

Cf. 4th column of A144064.

Sequence in context: A278680 A121593 A160527 * A001872 A054443 A281766

Adjacent sequences:  A023000 A023001 A023002 * A023004 A023005 A023006

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified March 22 22:17 EDT 2017. Contains 283901 sequences.