A382344 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 4 kinds.

1, 0, 4, 0, 4, 10, 0, 4, 16, 20, 0, 4, 26, 40, 35, 0, 4, 32, 80, 80, 56, 0, 4, 42, 124, 180, 140, 84, 0, 4, 48, 184, 320, 340, 224, 120, 0, 4, 58, 248, 535, 660, 574, 336, 165, 0, 4, 64, 332, 800, 1200, 1184, 896, 480, 220, 0, 4, 74, 416, 1176, 1956, 2284, 1932, 1320, 660, 286

Offset: 0

Author

Peter Dolland, Mar 28 2025

Keywords

• nonn
• tabl

Examples

			Triangle starts:
 0 : [1]
 1 : [0, 4]
 2 : [0, 4, 10]
 3 : [0, 4, 16,  20]
 4 : [0, 4, 26,  40,   35]
 5 : [0, 4, 32,  80,   80,   56]
 6 : [0, 4, 42, 124,  180,  140,   84]
 7 : [0, 4, 48, 184,  320,  340,  224,  120]
 8 : [0, 4, 58, 248,  535,  660,  574,  336,  165]
 9 : [0, 4, 64, 332,  800, 1200, 1184,  896,  480, 220]
10 : [0, 4, 74, 416, 1176, 1956, 2284, 1932, 1320, 660, 286]
...
		

Crossrefs

Main diagonal gives A000292(n+1).

Row sums give A023003.

Cf. A008284 (1-kind), A382342 (2-kind), A382343 (3-kind).

Cf. A022599, A382041.

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))*binomial(j+3, 3), 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 28 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]]*Binomial[j+3, 3], {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, Aug 07 2025, after Alois P. Heinz *)

Python

from sympy import binomial
from sympy.utilities.iterables import partitions
kinds = 4 - 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 + 3, 3) = A000292(n + 1).

T(n,1) = 4 for n >= 1.

T(n,k) = A382041(n,k) - A382041(n,k-1) for 1 <= k <= n.

Sum_{k=0..n} (-1)^k * T(n,k) = A022599(n). - Alois P. Heinz, Mar 28 2025