

A254811


Number of ways to put n red, n blue, and n green balls into n indistinguishable boxes.


3



1, 1, 14, 171, 1975, 20096, 187921, 1609727, 12827392, 95701382, 673873648, 4503935052, 28728268655, 175644353402, 1033386471872, 5870110651051, 32289704469531, 172438417419444, 896076816466546, 4540173176769827, 22469530730320361
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OFFSET

0,3


COMMENTS

a(n) is the sum of the number of partitions of the multiset {R^n, B^n, G^n} into 1, 2, ..., n parts (as observed in the pink box comments by Joerg Arndt and Tom Edgar). a(0) := 1. For partitions of multisets see the Knuth reference.  Wolfdieter Lang, Mar 26 2015
a(n) is also the number of factorizations of m^n into at most n factors where m is a product of 3 distinct primes. a(2) = 14: (2*3*5)^2 = 900 has 14 factorizations into at most 2 factors: 900, 30*30, 36*25, 45*20, 50*18, 60*15, 75*12, 90*10, 100*9, 150*6, 180*5, 225*4, 300*3, 450*2.  Alois P. Heinz, Mar 26 2015


REFERENCES

D. A. Knuth, The Art of Computer Programming. Volume 4, Fascicle 3, AddisonWesley, 2010, pp. 74  77.


LINKS

Brian Chen, Table of n, a(n) for n = 0..64


EXAMPLE

For n = 2 the a(2) = 14 ways to put 2 red balls, 2 blue balls, and 2 green balls into 2 indistinguishable boxes are (RRBBGG)(), (RRBBG)(G), (RRBGG)(B), (RBBGG)(R), (RRBB)(GG), (RRGG)(BB), (BBGG)(RR), (RRBG)(BG), (RBBG)(RG), (RBGG)(RB), (RRB)(BGG), (RBB)(RGG), (RRG)(BBG), (RGB)(RGB).


MAPLE

with(numtheory):
b:= proc(n, k, i) option remember;
`if`(n>k, 0, 1) +`if`(isprime(n) or i<2, 0, add(
`if`(d>k, 0, b(n/d, d, i1)), d=divisors(n) minus {1, n}))
end:
a:= n> b(30^n$2, n):
seq(a(n), n=0..8); # Alois P. Heinz, Mar 26 2015


MATHEMATICA

b[n_, k_, i_] := b[n, k, i] = If[n>k, 0, 1] + If[PrimeQ[n]  i<2, 0, Sum[ If[d>k, 0, b[n/d, d, i1]], {d, Divisors[n][[2 ;; 2]]}]]; a[n_] := b[30^n, 30^n, n]; Table[a[n], {n, 0, 8}] (* JeanFrançois Alcover, Jan 08 2016, after Alois P. Heinz *)


CROSSREFS

Cf. A254686.
Column k=3 of A256384.
Sequence in context: A199529 A098299 A341500 * A099158 A014882 A048443
Adjacent sequences: A254808 A254809 A254810 * A254812 A254813 A254814


KEYWORD

nonn


AUTHOR

Brian Chen, Feb 08 2015


STATUS

approved



