A383351 Triangle read by rows: T(n, k) is the number of partitions of a 2-colored set of n objects into k parts where 0 <= k <= n, and each part is one of 2 kinds.

1, 0, 4, 0, 6, 10, 0, 8, 24, 20, 0, 10, 53, 60, 35, 0, 12, 88, 164, 120, 56, 0, 14, 144, 348, 370, 210, 84, 0, 16, 208, 672, 904, 700, 336, 120, 0, 18, 299, 1174, 1998, 1870, 1183, 504, 165, 0, 20, 400, 1952, 3952, 4524, 3360, 1848, 720, 220

Offset: 0

Author

Peter Dolland, Apr 24 2025

Keywords

• nonn
• tabl

Examples

			Triangle starts:
 0 : [1]
 1 : [0,  4]
 2 : [0,  6,  10]
 3 : [0,  8,  24,   20]
 4 : [0, 10,  53,   60,   35]
 5 : [0, 12,  88,  164,  120,   56]
 6 : [0, 14, 144,  348,  370,  210,   84]
 7 : [0, 16, 208,  672,  904,  700,  336,  120]
 8 : [0, 18, 299, 1174, 1998, 1870, 1183,  504,  165]
 9 : [0, 20, 400, 1952, 3952, 4524, 3360, 1848,  720, 220]
10 : [0, 22, 534, 3052, 7394, 9834, 8652, 5488, 2724, 990, 286]
...
		

Crossrefs

Main diagonal gives A000292(n+1).

Partial row sums are A383352.

Cf. A382342 (1-colored), A382339 (1-kind), A008284 (1-colored, 1-kind).

Programs

Python

from sympy import binomial
from sympy.utilities.iterables import partitions
def calc_w(k , m):
    s = 0
    for p in partitions(m, m=k+1):
        fact = 1
        j = k + 1
        for x in p :
            fact *= binomial(j, p[x]) * (x + 1) ** p[x]
            j -= p[x]
        s += fact
    return s
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 *= calc_w(k, p[k])
        if s > 0 :
            t[s - 1] += fact
    return [0] + t

Formula

T(n,n) = binomial(n + 3, 3) = A000292(n + 1).

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

T(n,k+1) = A383352(n,k+1) - A383352(n,k) for 0 <= k < n.