A383352 Triangle read by rows: T(n, k) is the number of partitions of a 2-colored set of n objects into at most k parts where 0 <= k <= n, and each part is one of 2 kinds.

1, 0, 4, 0, 6, 16, 0, 8, 32, 52, 0, 10, 63, 123, 158, 0, 12, 100, 264, 384, 440, 0, 14, 158, 506, 876, 1086, 1170, 0, 16, 224, 896, 1800, 2500, 2836, 2956, 0, 18, 317, 1491, 3489, 5359, 6542, 7046, 7211, 0, 20, 420, 2372, 6324, 10848, 14208, 16056, 16776, 16996

Offset: 0

Author

Peter Dolland, Apr 24 2025

Keywords

• nonn
• tabl

Examples

			Triangle starts:
 0 : [1]
 1 : [0,  4]
 2 : [0,  6,  16]
 3 : [0,  8,  32,   52]
 4 : [0, 10,  63,  123,   158]
 5 : [0, 12, 100,  264,   384,   440]
 6 : [0, 14, 158,  506,   876,  1086,  1170]
 7 : [0, 16, 224,  896,  1800,  2500,  2836,  2956]
 8 : [0, 18, 317, 1491,  3489,  5359,  6542,  7046,  7211]
 9 : [0, 20, 420, 2372,  6324, 10848, 14208, 16056, 16776, 16996]
10 : [0, 22, 556, 3608, 11002, 20836, 29488, 34976, 37700, 38690, 38976]
...
		

Crossrefs

Row sums of A383351.

Cf. A381895 (1-color), A381891 (1-kind), A026820 (1-color, 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
    for i in range(n - 1):
        t[i + 1] += t[i]
    return [0] + t

Formula

T(n,k) = Sum_{i=0..k} A383351(n,i).

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