A256384 -id:A256384 - cp's OEIS frontend

A219727 Number A(n,k) of k-partite partitions of {n}^k into k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 9, 3, 1, 1, 15, 66, 31, 5, 1, 1, 52, 712, 686, 109, 7, 1, 1, 203, 10457, 27036, 6721, 339, 11, 1, 1, 877, 198091, 1688360, 911838, 58616, 1043, 15, 1, 1, 4140, 4659138, 154703688, 231575143, 26908756, 476781, 2998, 22, 1

Offset: 0

Alois P. Heinz, Nov 26 2012

A(n,k) is the number of factorizations of m^n where m is a product of k distinct primes. A(2,2) = 9: (2*3)^2 = 36 has 9 factorizations: 36, 3*12, 4*9, 3*3*4, 2*18, 6*6, 2*3*6, 2*2*9, 2*2*3*3.

A(n,k) is the number of (n*k) X k matrices with nonnegative integer entries and column sums n up to permutation of rows. - Andrew Howroyd, Dec 10 2018

			A(1,3) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(1,0,0),(0,1,0),(0,0,1)], [(0,1,1),(1,0,0)].
A(2,2) = 9: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(2,0),(0,1),(0,1)], [(1,2),(1,0)], [(1,1),(1,1)], [(1,1),(1,0),(0,1)], [(1,0),(1,0),(0,2)], [(1,0),(1,0),(0,1),(0,1)].
Square array A(n,k) begins:
  1,   1,    1,      1,        1,         1,         1,       1, ...
  1,   1,    2,      5,       15,        52,       203,     877, ...
  1,   2,    9,     66,      712,     10457,    198091, 4659138, ...
  1,   3,   31,    686,    27036,   1688360, 154703688, ...
  1,   5,  109,   6721,   911838, 231575143, ...
  1,   7,  339,  58616, 26908756, ...
  1,  11, 1043, 476781, ...
  1,  15, 2998, ...

Andrew Howroyd, Table of n, a(n) for n = 0..209

Columns k=0..3 give: A000012, A000041, A002774, A219678.
Rows n=0..4 give: A000012, A000110, A020555, A322487, A358781.
Main diagonal gives A322488.
Cf. A188392, A219585 (partitions of {n}^k into distinct k-tuples), A256384, A318284, A318951.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); EulerT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, 1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p,n,k), [1,n]); s*q[#q-j]))} \\ Andrew Howroyd, Dec 11 2018

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 *)

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

Original entry on oeis.org

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 *)

A256493 Number of factorizations of m^3 into at most 3 factors, where m is a product of exactly n distinct primes.

Original entry on oeis.org

1, 3, 19, 171, 1675, 16683, 166699, 1666731, 16666795, 166666923, 1666667179, 16666667691, 166666668715, 1666666670763, 16666666674859, 166666666683051, 1666666666699435, 16666666666732203, 166666666666797739, 1666666666666928811, 16666666666667190955

Offset: 0

Alois P. Heinz, Mar 30 2015

Also the number of n-partite partitions of (3)^n into at most 3 n-tuples.

			The a(1) = 3 factorizations of 2^3 into at most 3 factors are: 8, 2*4, 2*2*2.
The a(2) = 19 factorizations of (2*3)^3 into at most 3 factors are: 216, 2*108, 3*72, 4*54, 6*36, 8*27, 9*24, 12*18, 2*2*54, 2*3*36, 2*4*27, 2*6*18, 2*9*12, 3*3*24, 3*4*18, 3*6*12, 3*8*9, 4*6*9, 6*6*6.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-32,20).

Row n=3 of A256384.

a:= n-> (10^n + 3*2^n + 2)/6: seq(a(n), n=0..30);

LinearRecurrence[{13,-32,20},{1,3,19},30] (* Harvey P. Dale, Dec 30 2019 *)

G.f.: -(12*x^2-10*x+1)/((x-1)*(2*x-1)*(10*x-1)).

a(n) = (10^n + 3*2^n + 2)/6.

cp's OEIS Frontend

A219727 Number A(n,k) of k-partite partitions of {n}^k into k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

A256493 Number of factorizations of m^3 into at most 3 factors, where m is a product of exactly n distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula