A023006 Number of partitions of n into parts of 7 kinds.

1, 7, 35, 140, 490, 1547, 4522, 12405, 32305, 80465, 192899, 447146, 1006145, 2204475, 4715510, 9869132, 20247710, 40786690, 80782800, 157510780, 302666903, 573720808, 1073720305, 1985506775, 3630307835, 6567206471, 11760658378, 20860415590, 36665885170, 63891010155, 110415782785, 189320804673, 322174588225

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is Euler transform of A010727. - Alois P. Heinz, Oct 17 2008

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)
• Roland Bacher, P. De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016, hal-01285685v2.
• P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, 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

Cf. 7th column of A144064. - Alois P. Heinz, Oct 17 2008

Programs

Maple

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*7, 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)^7,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

PARI

Vec(1/eta('x+O('x^66))^7)  /* Joerg Arndt, Jul 30 2011 */

Formula

G.f.: Product_{m>=1} 1/(1-x^m)^7.

a(n) ~ 49 * exp(Pi * sqrt(14*n/3)) / (576 * sqrt(2) * n^(5/2)). - Vaclav Kotesovec, Feb 28 2015

a(0) = 1, a(n) = (7/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017

G.f.: exp(7*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018