A382041 Triangle read by rows: T(n, k) is the number of partitions of n with at most k parts where 0 <= k <= n, and each part is one of four kinds.

1, 0, 4, 0, 4, 14, 0, 4, 20, 40, 0, 4, 30, 70, 105, 0, 4, 36, 116, 196, 252, 0, 4, 46, 170, 350, 490, 574, 0, 4, 52, 236, 556, 896, 1120, 1240, 0, 4, 62, 310, 845, 1505, 2079, 2415, 2580, 0, 4, 68, 400, 1200, 2400, 3584, 4480, 4960, 5180, 0, 4, 78, 494, 1670, 3626, 5910, 7842, 9162, 9822, 10108

Offset

0,3

Comments

Two unrestricted unary predicates on the parts set result in four kinds: The intersection, the both differences and the complement of the union.

The 1-kind case is Euler's table A026820.

The 2-kind case is A381895.

The 3-kind case is A382025.

Links

Table of n, a(n) for n=0..65.

Example

Triangle starts:
  0 : [1]
  1 : [0, 4]
  2 : [0, 4, 14]
  3 : [0, 4, 20,  40]
  4 : [0, 4, 30,  70,  105]
  5 : [0, 4, 36, 116,  196,  252]
  6 : [0, 4, 46, 170,  350,  490,  574]
  7 : [0, 4, 52, 236,  556,  896, 1120, 1240]
  8 : [0, 4, 62, 310,  845, 1505, 2079, 2415, 2580]
  9 : [0, 4, 68, 400, 1200, 2400, 3584, 4480, 4960, 5180]
 10 : [0, 4, 78, 494, 1670, 3626, 5910, 7842, 9162, 9822, 10108]
 ...

Prog

(Python)
from sympy import binomial
from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
kinds = 4 - 1   # the number of part kinds - 1
def a382041_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        p = IntegerPartition( p).as_dict()
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( kinds + p[k], kinds)
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t

Crossrefs

Main diagonal gives A023003.

Cf. A026820, A381895, A382025.

Sequence in context: A058536 A371377 A154854 * A151672 A058493 A112149

Adjacent sequences: A382038 A382039 A382040 * A382042 A382043 A382044

Keyword

nonn,tabl

Author

Peter Dolland, Mar 12 2025

Status

approved