A217093 Number of partitions of n objects of 3 colors.

1, 3, 12, 38, 117, 330, 906, 2367, 6027, 14873, 35892, 84657, 196018, 445746, 997962, 2201438, 4792005, 10300950, 21889368, 46012119, 95746284, 197344937, 403121547, 816501180, 1640549317, 3271188702, 6475456896, 12730032791, 24861111315, 48246729411, 93065426256

Offset

0,2

Comments

a(n) is also the number of unlabeled simple graphs with n nodes of 3 colors whose components are complete graphs.

Number of (integer) partitions of n into 3 sorts of part 1, 6 sorts of part 2, 10 sorts of part 3, ..., (k+2)*(k+1)/2 sorts of part k. - Joerg Arndt, Dec 07 2014

In general the g.f. 1 / prod(n>=1, (1-x^k)^m(k) ) gives the number of (integer) partitions where there are m(k) sorts of part k. - Joerg Arndt, Mar 10 2015

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

S. Benvenuti, B. Feng, A. Hanany and Y. H. He, Counting BPS operators in gauge theories: Quivers, syzygies and plethystics, arXiv:hep-th/0608050, Aug 2006, p.42.

Carlos A. A. Florentino, Plethystic exponential calculus and permutation polynomials, arXiv:2105.13049 [math.CO], 2021. Mentions this sequence.

Vaclav Kotesovec, Graph - The asymptotic ratio

Formula

G.f.: Product_{i>=1} 1/(1-x^i)^binomial(i+2,2).

EULER transform of 3, 6, 10, 15, ... .

Generally for the number of partitions of k colors the generating function is Product_{i>=1} 1/(1-x^i)^binomial(i+k-1,k-1).

a(n) ~ Pi^(1/8) * exp(1/8 + 3^4 * 5^2 * Zeta(3)^3 / (2*Pi^8) - 31*Zeta(3) / (8*Pi^2) + 5^(1/4) * Pi * n^(1/4) / 6^(3/4) - 3^(13/4) * 5^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(7/4) * Pi^5) + 3^(3/2) * 5^(1/2) * Zeta(3) * n^(1/2) / (2^(1/2) * Pi^2) + 2^(7/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4))) / (A^(3/2) * 2^(73/32) * 15^(9/32) * n^(25/32)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 08 2015

Example

We represent each summand, k, in a partition of n as k identical objects. Then we color each object. We have no regard for the order of the colored objects.
a(2) = 12 because we have: ww; wg; wb; gg; gb; bb; w + w; w + g; w + b; g + g; g + b; b + b, where the 3 colors are white w, gray g, and black b.

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*binomial(d+2, 2), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 26 2012

Mathematica

nn=30; p=Product[1/(1- x^i)^Binomial[i+2, 2], {i, 1, nn}]; CoefficientList[Series[p, {x, 0, nn}], x]

Crossrefs

Cf. A005380, A255050, A255052.

Column k=3 of A075196.

Sequence in context: A008907 A048246 A320203 * A278639 A350787 A222643

Adjacent sequences: A217090 A217091 A217092 * A217094 A217095 A217096

Keyword

nonn

Author

Geoffrey Critzer, Sep 26 2012

Status

approved