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 A217093 Number of partitions of n objects of 3 colors. 7
 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 (list; graph; refs; listen; history; text; internal format)
 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. 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 A222643 A129014 Adjacent sequences:  A217090 A217091 A217092 * A217094 A217095 A217096 KEYWORD nonn AUTHOR Geoffrey Critzer, Sep 26 2012 STATUS approved

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Last modified July 12 23:28 EDT 2020. Contains 335669 sequences. (Running on oeis4.)