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A382342 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 two kinds.

%I A382342 #22 Apr 19 2025 03:53:53
%S A382342 1,0,2,0,2,3,0,2,4,4,0,2,7,6,5,0,2,8,12,8,6,0,2,11,18,17,10,7,0,2,12,
%T A382342 26,28,22,12,8,0,2,15,34,46,38,27,14,9,0,2,16,46,64,66,48,32,16,10,0,
%U A382342 2,19,56,94,100,86,58,37,18,11,0,2,20,70,124,152,136,106,68,42,20,12
%N A382342 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 two kinds.
%H A382342 Alois P. Heinz, <a href="/A382342/b382342.txt">Rows n = 0..200, flattened</a>
%F A382342 T(n,n) = n + 1.
%F A382342 T(n,1) = 2 for n >= 1.
%F A382342 T(n,k) = A381895(n,k) - A381895(n,k-1) for 1 <= k <= n.
%F A382342 Sum_{k=0..n} (-1)^k * T(n,k) = A022597(n). - _Alois P. Heinz_, Mar 27 2025
%e A382342 Triangle starts:
%e A382342  0 : [1]
%e A382342  1 : [0, 2]
%e A382342  2 : [0, 2,  3]
%e A382342  3 : [0, 2,  4,  4]
%e A382342  4 : [0, 2,  7,  6,  5]
%e A382342  5 : [0, 2,  8, 12,  8,   6]
%e A382342  6 : [0, 2, 11, 18, 17,  10,  7]
%e A382342  7 : [0, 2, 12, 26, 28,  22, 12,  8]
%e A382342  8 : [0, 2, 15, 34, 46,  38, 27, 14,  9]
%e A382342  9 : [0, 2, 16, 46, 64,  66, 48, 32, 16, 10]
%e A382342 10 : [0, 2, 19, 56, 94, 100, 86, 58, 37, 18, 11]
%e A382342   ...
%p A382342 b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
%p A382342       add(x^j*b(n-i*j, min(n-i*j, i-1))*(j+1), j=0..n/i))))
%p A382342     end:
%p A382342 T:= (n, k)-> coeff(b(n$2), x, k):
%p A382342 seq(seq(T(n, k), k=0..n), n=0..11);  # _Alois P. Heinz_, Mar 27 2025
%t A382342 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 + 1), {j, 0, n/i}]]]];
%t A382342 T[n_, k_] := Coefficient[b[n, n], x, k];
%t A382342 Table[Table[T[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, Apr 19 2025, after _Alois P. Heinz_ *)
%o A382342 (Python)
%o A382342 from sympy.utilities.iterables import partitions
%o A382342 def t_row( n):
%o A382342     if n == 0 : return [1]
%o A382342     t = list( [0] * n)
%o A382342     for p in partitions( n):
%o A382342         fact = 1
%o A382342         s = 0
%o A382342         for k in p :
%o A382342             s += p[k]
%o A382342             fact *= 1 + p[k]
%o A382342         if s > 0 :
%o A382342             t[s - 1] += fact
%o A382342     return [0] + t
%Y A382342 Row sums give A000712.
%Y A382342 Cf. A008284 (1-kind case), A022597, A381895, A382345.
%K A382342 nonn,tabl
%O A382342 0,3
%A A382342 _Peter Dolland_, Mar 27 2025