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A254811
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Number of ways to put n red, n blue, and n green balls into n indistinguishable boxes.
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3
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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
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0,3
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COMMENTS
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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
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REFERENCES
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D. A. Knuth, The Art of Computer Programming. Volume 4, Fascicle 3, Addison-Wesley, 2010, pp. 74 - 77.
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LINKS
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Brian Chen, Table of n, a(n) for n = 0..64
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EXAMPLE
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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).
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MAPLE
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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, i-1)), 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
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MATHEMATICA
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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, i-1]], {d, Divisors[n][[2 ;; -2]]}]]; a[n_] := b[30^n, 30^n, n]; Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)
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CROSSREFS
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Cf. A254686.
Column k=3 of A256384.
Sequence in context: A199529 A098299 A341500 * A099158 A014882 A048443
Adjacent sequences: A254808 A254809 A254810 * A254812 A254813 A254814
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KEYWORD
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nonn
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AUTHOR
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Brian Chen, Feb 08 2015
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STATUS
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approved
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