A382340 Triangle read by rows: T(n,k) is the number of partitions of a 3-colored set of n objects into exactly k parts with 0 <= k <= n.

1, 0, 3, 0, 6, 6, 0, 10, 18, 10, 0, 15, 51, 36, 15, 0, 21, 105, 123, 60, 21, 0, 28, 208, 326, 226, 90, 28, 0, 36, 360, 771, 678, 360, 126, 36, 0, 45, 606, 1641, 1836, 1161, 525, 168, 45, 0, 55, 946, 3271, 4431, 3403, 1775, 721, 216, 55, 0, 66, 1446, 6096, 10026, 8982, 5472, 2520, 948, 270, 66

Offset: 0

Author

Peter Dolland, Mar 22 2025

Keywords

• nonn
• tabl

Examples

			Triangle starts:
 0 : [1]
 1 : [0,  3]
 2 : [0,  6,    6]
 3 : [0, 10,   18,   10]
 4 : [0, 15,   51,   36,    15]
 5 : [0, 21,  105,  123,    60,   21]
 6 : [0, 28,  208,  326,   226,   90,   28]
 7 : [0, 36,  360,  771,   678,  360,  126,   36]
 8 : [0, 45,  606, 1641,  1836, 1161,  525,  168,  45]
 9 : [0, 55,  946, 3271,  4431, 3403, 1775,  721, 216,  55]
10 : [0, 66, 1446, 6096, 10026, 8982, 5472, 2520, 948, 270, 66]
...
		

Crossrefs

Row sums are A217093.

The 1-color case is A008284.

The 2-color case is A382339.

Cf. A382045.

Programs

Maple

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1))*binomial(i*(i+3)/2+j, j)*x^j, j=0..n/i))))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 22 2025

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1]]*Binomial[i*(i + 3)/2 + j, j]*x^j, {j, 0, n/i}]]]];
T[n_, k_] := Coefficient[b[n, n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 17 2025, after Alois P. Heinz *)

Python

from sympy import binomial
from sympy.utilities.iterables import partitions
colors = 3 - 1   # the number of colors - 1
def t_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( binomial( k + colors, colors) + p[k] - 1, p[k])
        if s > 0 :
            t[s - 1] += fact
    return [0] + t

Formula

T(n,1) = binomial(n + 2, 2) = A000217(n + 1) for n >= 1.

T(n,n) = binomial(n + 2, 2) = A000217(n + 1).

T(n,k) = A382045(n,k) - A382045(n,k-1) for k >= 1.