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%I #30 Nov 17 2023 11:57:00
%S 1,0,1,0,1,0,1,2,0,1,2,1,0,1,4,2,0,1,4,6,1,0,1,6,8,4,0,1,6,13,9,1,0,1,
%T 8,18,16,6,0,1,8,24,29,13,2,0,1,10,30,43,29,6,0,1,10,39,64,52,19,1,0,
%U 1,12,46,89,89,42,7,0,1,12,56,122,139,85,22,1
%N Number T(n,k) of sets of nonempty strict integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=max(i:T(n,i)>0), read by rows.
%C T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.
%H Alois P. Heinz, <a href="/A360764/b360764.txt">Rows n = 0..250, flattened</a>
%e T(6,1) = 1: {[6]}.
%e T(6,2) = 4: {[1],[5]}, {[2],[4]}, {[1,5]}, {[2,4]}.
%e T(6,3) = 6: {[1,2,3]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[3],[1,2]}, {[1],[2],[3]}.
%e T(6,4) = 1: {[1],[2],[1,2]}.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1;
%e 0, 1, 2;
%e 0, 1, 2, 1;
%e 0, 1, 4, 2;
%e 0, 1, 4, 6, 1;
%e 0, 1, 6, 8, 4;
%e 0, 1, 6, 13, 9, 1;
%e 0, 1, 8, 18, 16, 6;
%e 0, 1, 8, 24, 29, 13, 2;
%e 0, 1, 10, 30, 43, 29, 6;
%e 0, 1, 10, 39, 64, 52, 19, 1;
%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-1)))))
%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-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p seq(T(n), n=0..14);
%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 - 1]]]]];
%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_] := CoefficientList[b[n, n], x];
%t Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Nov 17 2023, after _Alois P. Heinz_ *)
%Y Columns k=0-2 give: A000007, A057427, A052928(n-1) for n>=3.
%Y Row sums give A050342.
%Y Cf. A000009, A008289, A055884, A330462, A360742, A360763.
%K nonn,tabf
%O 0,8
%A _Alois P. Heinz_, Feb 19 2023
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