A382343 Triangle read by rows: T(n, k) is the number of partitions of n into k parts where 0 <= k <= n, and each part is one of 3 kinds.

1, 0, 3, 0, 3, 6, 0, 3, 9, 10, 0, 3, 15, 18, 15, 0, 3, 18, 36, 30, 21, 0, 3, 24, 55, 66, 45, 28, 0, 3, 27, 81, 114, 105, 63, 36, 0, 3, 33, 108, 189, 195, 153, 84, 45, 0, 3, 36, 145, 276, 348, 298, 210, 108, 55, 0, 3, 42, 180, 405, 552, 558, 423, 276, 135, 66

Offset: 0

Author

Peter Dolland, Mar 27 2025

Keywords

• nonn
• tabl

Examples

			Triangle starts:
 0 : [1]
 1 : [0, 3]
 2 : [0, 3,  6]
 3 : [0, 3,  9,  10]
 4 : [0, 3, 15,  18,  15]
 5 : [0, 3, 18,  36,  30,  21]
 6 : [0, 3, 24,  55,  66,  45,  28]
 7 : [0, 3, 27,  81, 114, 105,  63,  36]
 8 : [0, 3, 33, 108, 189, 195, 153,  84,  45]
 9 : [0, 3, 36, 145, 276, 348, 298, 210, 108,  55]
10 : [0, 3, 42, 180, 405, 552, 558, 423, 276, 135, 66]
...
		

Crossrefs

Main diagonal gives A000217(n+1).

Row sums give A000716.

Cf. A008284 (1-kind), A382342 (2-kind).

Cf. A022598, A382025.

Programs

Maple

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(x^j*b(n-i*j, min(n-i*j, i-1))*(j+2)*(j+1)/2, 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 27 2025

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^j*b[n-i*j, Min[n-i*j, i-1]]*(j+2)*(j+1)/2, {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, Jul 30 2025, after Alois P. Heinz *)

Python

from sympy import binomial
from sympy.utilities.iterables import partitions
kinds = 3 - 1   # the number of part kinds - 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( kinds + p[k], kinds)
        if s > 0 :
            t[s - 1] += fact
    return [0] + t

Formula

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

T(n,1) = 3 for n >= 1.

T(n,k) = A382025(n,k) - A382025(n,k-1) for 1 <= k <= n.

Sum_{k=0..n} (-1)^k * T(n,k) = A022598(n). - Alois P. Heinz, Mar 27 2025