Brian Chen - cp's OEIS frontend

User: Brian Chen

Brian Chen has authored 2 sequences.

A254811 Number of ways to put n red, n blue, and n green balls into n indistinguishable boxes.

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

Offset: 0

Brian Chen, Feb 08 2015

nonn

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

			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).

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

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

Cf. A254686.

Column k=3 of A256384.

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

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 *)

A254686 Number of ways to put n red and n blue balls into n indistinguishable boxes.

Original entry on oeis.org

1, 1, 5, 19, 74, 248, 814, 2457, 7168, 19928, 53688, 139820, 354987, 878434, 2128102, 5052010, 11781881, 27019758, 61035671, 135928105, 298784144, 648726349, 1392474574, 2956730910, 6214668074, 12937060340, 26686392239, 54572423946, 110680119454, 222710856175, 444776676764

Offset: 0

Brian Chen, Feb 08 2015

nonn

See a comment on A254811 about multiset partitions and the Knuth reference. - Wolfdieter Lang, Mar 26 2015

			For n = 2 the a(2) = 5 ways to put 2 red balls and 2 blue balls into 2 indistinguishable boxes are (RRBB)(), (RRB)(B), (RBB)(R), (RR)(BB), (RB)(RB).

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

Cf. A002774, A060150, A254811.
Column k=2 of A256384.
Main diagonal of A277239.

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(6^n$2,n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 26 2015

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[6^n, 6^n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

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User: Brian Chen

A254811 Number of ways to put n red, n blue, and n green balls into n indistinguishable boxes.

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