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%I #28 Nov 15 2023 08:09:30
%S 1,0,1,0,1,1,0,1,2,2,0,1,3,3,2,0,1,4,6,5,3,0,1,5,10,10,7,4,0,1,6,14,
%T 19,16,10,5,0,1,7,19,30,32,24,14,6,0,1,8,26,46,57,52,35,19,8,0,1,9,32,
%U 67,94,97,79,50,25,10,0,1,10,40,93,147,172,157,117,69,33,12
%N Number T(n,k) of sets of nonempty integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A360742/b360742.txt">Rows n = 0..200, flattened</a>
%F T(n,n) + T(n+1,n) = T(n+2,n+1) for n>=0.
%e T(6,3) = 10: {[1,1,4]}, {[1,2,3]}, {[2,2,2]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[2],[2,2]}, {[3],[1,2]}, {[4],[1,1]}, {[1],[2],[3]}.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 2, 2;
%e 0, 1, 3, 3, 2;
%e 0, 1, 4, 6, 5, 3;
%e 0, 1, 5, 10, 10, 7, 4;
%e 0, 1, 6, 14, 19, 16, 10, 5;
%e 0, 1, 7, 19, 30, 32, 24, 14, 6;
%e 0, 1, 8, 26, 46, 57, 52, 35, 19, 8;
%e 0, 1, 9, 32, 67, 94, 97, 79, 50, 25, 10;
%e ...
%p h:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i)))))
%p end:
%p g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
%p g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i), k), k=0..j))))
%p end:
%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
%p end:
%p T:= (n, k)-> coeff(b(n$2), x, k):
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t h[n_, i_] := h[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, h[n, i - 1] + x*h[n - i, Min[n - i, i]]]]];
%t g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[ g[n, i - 1, j - k]*x^(i*k)*Binomial[Coefficient[h[n, n], x, i], k], {k, 0, j}]]]];
%t b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
%t T[n_, k_] := Coefficient[b[n, n], x, k];
%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Nov 15 2023, after _Alois P. Heinz_ *)
%Y Columns k=0-2 give: A000007, A057427, A001477(n-1) for n>=1.
%Y Main diagonal gives A000009.
%Y T(n+2,n+1) gives A036469.
%Y Row sums give A261049.
%Y T(2n,n) gives A360714.
%Y Cf. A000041, A055884 (similar triangle for multisets), A330463.
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, Feb 18 2023
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