A217093 - OEIS

%I #57 Jan 16 2025 19:37:38

%S 1,3,12,38,117,330,906,2367,6027,14873,35892,84657,196018,445746,

%T 997962,2201438,4792005,10300950,21889368,46012119,95746284,197344937,

%U 403121547,816501180,1640549317,3271188702,6475456896,12730032791,24861111315,48246729411,93065426256

%N Number of partitions of n objects of 3 colors.

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

%C 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

%C 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

%H Alois P. Heinz, <a href="/A217093/b217093.txt">Table of n, a(n) for n = 0..1000</a>

%H S. Benvenuti, B. Feng, A. Hanany and Y. H. He, <a href="https://arxiv.org/abs/hep-th/0608050">Counting BPS operators in gauge theories: Quivers, syzygies and plethystics</a>, arXiv:hep-th/0608050, Aug 2006, p.42.

%H Carlos A. A. Florentino, <a href="https://arxiv.org/abs/2105.13049">Plethystic exponential calculus and permutation polynomials</a>, arXiv:2105.13049 [math.CO], 2021. Mentions this sequence.

%H Vaclav Kotesovec, <a href="/A217093/a217093.jpg">Graph - The asymptotic ratio</a>

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

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

%F 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).

%F 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

%F G.f.: exp(Sum_{k >= 1} ((1/2)*sigma_3(k) + (3/2)*sigma_2(k) + sigma_1(k))*x^k/k) = 1 + 3*x + 12*x^2 + 38*x^3 + 117*x^4 + .... - _Peter Bala_, Jan 16 2025

%e 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.

%e 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.

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(

%p       d*binomial(d+2, 2), d=divisors(j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 26 2012

%p with(numtheory):

%p series(exp(add(((1/2)*sigma[3](k) + (3/2)*sigma[2](k) + sigma[1](k))*x^k/k, k = 1..30)), x, 31):

%p seq(coeftayl(%, x = 0, n), n = 0..30); # _Peter Bala_, Jan 16 2025

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

%Y Cf. A005380, A120844, A253289, A255050, A255052, A255802, A255803, A255835, A255836, A363601, A363602.

%Y Column k=3 of A075196.

%K nonn,easy

%O 0,2

%A _Geoffrey Critzer_, Sep 26 2012