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A382343 Triangle read by rows: T(n, k) is the number of partitions of n into k parts where 0 <= k <= n, and each part is one of 3 kinds.

%I A382343 #19 Aug 01 2025 01:57:25
%S A382343 1,0,3,0,3,6,0,3,9,10,0,3,15,18,15,0,3,18,36,30,21,0,3,24,55,66,45,28,
%T A382343 0,3,27,81,114,105,63,36,0,3,33,108,189,195,153,84,45,0,3,36,145,276,
%U A382343 348,298,210,108,55,0,3,42,180,405,552,558,423,276,135,66
%N A382343 Triangle read by rows: T(n, k) is the number of partitions of n into k parts where 0 <= k <= n, and each part is one of 3 kinds.
%F A382343 T(n,n) = binomial(n + 2, 2) = A000217(n + 1).
%F A382343 T(n,1) = 3 for n >= 1.
%F A382343 T(n,k) = A382025(n,k) - A382025(n,k-1) for 1 <= k <= n.
%F A382343 Sum_{k=0..n} (-1)^k * T(n,k) = A022598(n). - _Alois P. Heinz_, Mar 27 2025
%e A382343 Triangle starts:
%e A382343  0 : [1]
%e A382343  1 : [0, 3]
%e A382343  2 : [0, 3,  6]
%e A382343  3 : [0, 3,  9,  10]
%e A382343  4 : [0, 3, 15,  18,  15]
%e A382343  5 : [0, 3, 18,  36,  30,  21]
%e A382343  6 : [0, 3, 24,  55,  66,  45,  28]
%e A382343  7 : [0, 3, 27,  81, 114, 105,  63,  36]
%e A382343  8 : [0, 3, 33, 108, 189, 195, 153,  84,  45]
%e A382343  9 : [0, 3, 36, 145, 276, 348, 298, 210, 108,  55]
%e A382343 10 : [0, 3, 42, 180, 405, 552, 558, 423, 276, 135, 66]
%e A382343 ...
%p A382343 b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
%p A382343       add(x^j*b(n-i*j, min(n-i*j, i-1))*(j+2)*(j+1)/2, j=0..n/i))))
%p A382343     end:
%p A382343 T:= (n, k)-> coeff(b(n$2), x, k):
%p A382343 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Mar 27 2025
%t A382343 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^j*b[n-i*j, Min[n-i*j, i-1]]*(j+2)*(j+1)/2, {j, 0, n/i}]]]];
%t A382343 T[n_, k_] := Coefficient[b[n, n], x, k];
%t A382343 Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jul 30 2025, after _Alois P. Heinz_ *)
%o A382343 (Python)
%o A382343 from sympy import binomial
%o A382343 from sympy.utilities.iterables import partitions
%o A382343 kinds = 3 - 1   # the number of part kinds - 1
%o A382343 def t_row( n):
%o A382343     if n == 0 : return [1]
%o A382343     t = list( [0] * n)
%o A382343     for p in partitions( n):
%o A382343         fact = 1
%o A382343         s = 0
%o A382343         for k in p :
%o A382343             s += p[k]
%o A382343             fact *= binomial( kinds + p[k], kinds)
%o A382343         if s > 0 :
%o A382343             t[s - 1] += fact
%o A382343     return [0] + t
%Y A382343 Main diagonal gives A000217(n+1).
%Y A382343 Row sums give A000716.
%Y A382343 Cf. A008284 (1-kind), A382342 (2-kind).
%Y A382343 Cf. A022598, A382025.
%K A382343 nonn,tabl
%O A382343 0,3
%A A382343 _Peter Dolland_, Mar 27 2025