A382342 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 two kinds.

1, 0, 2, 0, 2, 3, 0, 2, 4, 4, 0, 2, 7, 6, 5, 0, 2, 8, 12, 8, 6, 0, 2, 11, 18, 17, 10, 7, 0, 2, 12, 26, 28, 22, 12, 8, 0, 2, 15, 34, 46, 38, 27, 14, 9, 0, 2, 16, 46, 64, 66, 48, 32, 16, 10, 0, 2, 19, 56, 94, 100, 86, 58, 37, 18, 11, 0, 2, 20, 70, 124, 152, 136, 106, 68, 42, 20, 12

Offset: 0

Author

Peter Dolland, Mar 27 2025

Keywords

• nonn
• tabl

Examples

			Triangle starts:
 0 : [1]
 1 : [0, 2]
 2 : [0, 2,  3]
 3 : [0, 2,  4,  4]
 4 : [0, 2,  7,  6,  5]
 5 : [0, 2,  8, 12,  8,   6]
 6 : [0, 2, 11, 18, 17,  10,  7]
 7 : [0, 2, 12, 26, 28,  22, 12,  8]
 8 : [0, 2, 15, 34, 46,  38, 27, 14,  9]
 9 : [0, 2, 16, 46, 64,  66, 48, 32, 16, 10]
10 : [0, 2, 19, 56, 94, 100, 86, 58, 37, 18, 11]
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Row sums give A000712.

Cf. A008284 (1-kind case), A022597, A381895, A382345.

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+1), j=0..n/i))))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..11);  # 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 + 1), {j, 0, n/i}]]]];
T[n_, k_] := Coefficient[b[n, n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Apr 19 2025, after Alois P. Heinz *)

Python

from sympy.utilities.iterables import partitions
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 *= 1 + p[k]
        if s > 0 :
            t[s - 1] += fact
    return [0] + t

Formula

T(n,n) = n + 1.

T(n,1) = 2 for n >= 1.

T(n,k) = A381895(n,k) - A381895(n,k-1) for 1 <= k <= n.

Sum_{k=0..n} (-1)^k * T(n,k) = A022597(n). - Alois P. Heinz, Mar 27 2025