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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.

%I A383352 #9 May 01 2025 18:41:31
%S A383352 1,0,4,0,6,16,0,8,32,52,0,10,63,123,158,0,12,100,264,384,440,0,14,158,
%T A383352 506,876,1086,1170,0,16,224,896,1800,2500,2836,2956,0,18,317,1491,
%U A383352 3489,5359,6542,7046,7211,0,20,420,2372,6324,10848,14208,16056,16776,16996
%N 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.
%F A383352 T(n,k) = Sum_{i=0..k} A383351(n,i).
%F A383352 T(n,1) = 2*n + 2 for n >= 1.
%e A383352 Triangle starts:
%e A383352  0 : [1]
%e A383352  1 : [0,  4]
%e A383352  2 : [0,  6,  16]
%e A383352  3 : [0,  8,  32,   52]
%e A383352  4 : [0, 10,  63,  123,   158]
%e A383352  5 : [0, 12, 100,  264,   384,   440]
%e A383352  6 : [0, 14, 158,  506,   876,  1086,  1170]
%e A383352  7 : [0, 16, 224,  896,  1800,  2500,  2836,  2956]
%e A383352  8 : [0, 18, 317, 1491,  3489,  5359,  6542,  7046,  7211]
%e A383352  9 : [0, 20, 420, 2372,  6324, 10848, 14208, 16056, 16776, 16996]
%e A383352 10 : [0, 22, 556, 3608, 11002, 20836, 29488, 34976, 37700, 38690, 38976]
%e A383352 ...
%o A383352 (Python)
%o A383352 from sympy import binomial
%o A383352 from sympy.utilities.iterables import partitions
%o A383352 def calc_w(k , m):
%o A383352     s = 0
%o A383352     for p in partitions(m, m=k+1):
%o A383352         fact = 1
%o A383352         j = k + 1
%o A383352         for x in p :
%o A383352             fact *= binomial(j, p[x]) * (x + 1) ** p[x]
%o A383352             j -= p[x]
%o A383352         s += fact
%o A383352     return s
%o A383352 def t_row(n):
%o A383352     if n == 0 : return [1]
%o A383352     t = list([0] * n)
%o A383352     for p in partitions( n):
%o A383352         fact = 1
%o A383352         s = 0
%o A383352         for k in p :
%o A383352             s += p[k]
%o A383352             fact *= calc_w(k, p[k])
%o A383352         if s > 0 :
%o A383352             t[s - 1] += fact
%o A383352     for i in range(n - 1):
%o A383352         t[i + 1] += t[i]
%o A383352     return [0] + t
%Y A383352 Row sums of A383351.
%Y A383352 Cf. A381895 (1-color), A381891 (1-kind), A026820 (1-color, 1-kind).
%K A383352 nonn,tabl
%O A383352 0,3
%A A383352 _Peter Dolland_, Apr 24 2025