A382041 - OEIS

%I #12 Mar 19 2025 10:39:55

%S 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,

%T 350,490,574,0,4,52,236,556,896,1120,1240,0,4,62,310,845,1505,2079,

%U 2415,2580,0,4,68,400,1200,2400,3584,4480,4960,5180,0,4,78,494,1670,3626,5910,7842,9162,9822,10108

%N 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.

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

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

%C The 2-kind case is A381895.

%C The 3-kind case is A382025.

%e Triangle starts:

%e   0 : [1]

%e   1 : [0, 4]

%e   2 : [0, 4, 14]

%e   3 : [0, 4, 20,  40]

%e   4 : [0, 4, 30,  70,  105]

%e   5 : [0, 4, 36, 116,  196,  252]

%e   6 : [0, 4, 46, 170,  350,  490,  574]

%e   7 : [0, 4, 52, 236,  556,  896, 1120, 1240]

%e   8 : [0, 4, 62, 310,  845, 1505, 2079, 2415, 2580]

%e   9 : [0, 4, 68, 400, 1200, 2400, 3584, 4480, 4960, 5180]

%e  10 : [0, 4, 78, 494, 1670, 3626, 5910, 7842, 9162, 9822, 10108]

%e  ...

%o (Python)

%o from sympy import binomial

%o from sympy.utilities.iterables import partitions

%o from sympy.combinatorics.partitions import IntegerPartition

%o kinds = 4 - 1   # the number of part kinds - 1

%o def a382041_row( n):

%o     if n == 0 : return [1]

%o     t = list( [0] * n)

%o     for p in partitions( n):

%o         p = IntegerPartition( p).as_dict()

%o         fact = 1

%o         s = 0

%o         for k in p :

%o             s += p[k]

%o             fact *= binomial( kinds + p[k], kinds)

%o         if s > 0 :

%o             t[s - 1] += fact

%o     for i in range( n - 1):

%o         t[i+1] += t[i]

%o     return [0] + t

%Y Main diagonal gives A023003.

%Y Cf. A026820, A381895, A382025.

%K nonn,tabl

%O 0,3

%A _Peter Dolland_, Mar 12 2025