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A382025 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 three kinds.

%I A382025 #5 Mar 19 2025 10:39:45
%S A382025 1,0,3,0,3,9,0,3,12,22,0,3,18,36,51,0,3,21,57,87,108,0,3,27,82,148,
%T A382025 193,221,0,3,30,111,225,330,393,429,0,3,36,144,333,528,681,765,810,0,
%U A382025 3,39,184,460,808,1106,1316,1424,1479,0,3,45,225,630,1182,1740,2163,2439,2574,2640
%N A382025 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 three kinds.
%C A382025 The 1-kind case is Euler's table A026820.
%C A382025 The 2-kind case is A381895.
%e A382025 Triangle starts:
%e A382025  0 : [1]
%e A382025  1 : [0, 3]
%e A382025  2 : [0, 3,  9]
%e A382025  3 : [0, 3, 12,  22]
%e A382025  4 : [0, 3, 18,  36,  51]
%e A382025  5 : [0, 3, 21,  57,  87,  108]
%e A382025  6 : [0, 3, 27,  82, 148,  193,  221]
%e A382025  7 : [0, 3, 30, 111, 225,  330,  393,  429]
%e A382025  8 : [0, 3, 36, 144, 333,  528,  681,  765,  810]
%e A382025  9 : [0, 3, 39, 184, 460,  808, 1106, 1316, 1424, 1479]
%e A382025 10 : [0, 3, 45, 225, 630, 1182, 1740, 2163, 2439, 2574, 2640]
%e A382025 ...
%o A382025 (Python)
%o A382025 from sympy import binomial
%o A382025 from sympy.utilities.iterables import partitions
%o A382025 from sympy.combinatorics.partitions import IntegerPartition
%o A382025 kinds = 3 - 1   # the number of part kinds - 1
%o A382025 def a382025_row( n):
%o A382025     if n == 0 : return [1]
%o A382025     t = list( [0] * n)
%o A382025     for p in partitions( n):
%o A382025         p = IntegerPartition( p).as_dict()
%o A382025         fact = 1
%o A382025         s = 0
%o A382025         for k in p :
%o A382025             s += p[k]
%o A382025             fact *= binomial( kinds + p[k], kinds)
%o A382025         if s > 0 :
%o A382025             t[s - 1] += fact
%o A382025     for i in range( n - 1):
%o A382025         t[i+1] += t[i]
%o A382025     return [0] + t
%Y A382025 Main diagonal gives A000716.
%Y A382025 Cf. A026820, A381895.
%K A382025 nonn,tabl
%O A382025 0,3
%A A382025 _Peter Dolland_, Mar 12 2025