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A360763 Number T(n,k) of multisets of nonempty strict 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.

%I A360763 #33 Sep 12 2023 12:02:46
%S A360763 1,0,1,0,1,1,0,1,2,1,0,1,3,2,1,0,1,4,4,2,1,0,1,5,8,5,2,1,0,1,6,11,10,
%T A360763 5,2,1,0,1,7,16,18,11,5,2,1,0,1,8,22,28,22,12,5,2,1,0,1,9,28,45,39,24,
%U A360763 12,5,2,1,0,1,10,35,63,67,46,25,12,5,2,1,0,1,11,44,89,106,86,50,26,12,5,2,1
%N A360763 Number T(n,k) of multisets of nonempty strict 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.
%C A360763 T(n,k) is defined for all n >= 0 and k >= 0.  Terms that are not in the triangle are zero.
%C A360763 Reversed rows and also the columns converge to A360785.
%H A360763 Alois P. Heinz, <a href="/A360763/b360763.txt">Rows n = 0..200, flattened</a>
%F A360763 T(3n,2n) = A360785(n) = T(3n+j,2n+j) for j>=0.
%e A360763 T(6,1) = 1: {[6]}.
%e A360763 T(6,2) = 5: {[1],[5]}, {[2],[4]}, {[3],[3]}, {[1,5]}, {[2,4]}.
%e A360763 T(6,3) = 8: {[1,2,3]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[3],[1,2]}, {[1],[1],[4]}, {[1],[2],[3]}, {[2],[2],[2]}.
%e A360763 T(6,4) = 5: {[1],[1],[1],[3]}, {[1],[1],[2],[2]}, {[1],[1],[1,3]}, {[1],[2],[1,2]}, {[1,2],[1,2]}.
%e A360763 T(6,5) = 2: {[1],[1],[1],[1],[2]}, {[1],[1],[1],[1,2]}.
%e A360763 T(6,6) = 1: {[1],[1],[1],[1],[1],[1]}.
%e A360763 Triangle T(n,k) begins:
%e A360763   1;
%e A360763   0, 1;
%e A360763   0, 1, 1;
%e A360763   0, 1, 2,  1;
%e A360763   0, 1, 3,  2,  1;
%e A360763   0, 1, 4,  4,  2,  1;
%e A360763   0, 1, 5,  8,  5,  2,  1;
%e A360763   0, 1, 6, 11, 10,  5,  2,  1;
%e A360763   0, 1, 7, 16, 18, 11,  5,  2, 1;
%e A360763   0, 1, 8, 22, 28, 22, 12,  5, 2, 1;
%e A360763   0, 1, 9, 28, 45, 39, 24, 12, 5, 2, 1;
%e A360763   ...
%p A360763 h:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A360763       `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i-1)))))
%p A360763     end:
%p A360763 g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
%p A360763       g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
%p A360763     end:
%p A360763 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A360763      `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
%p A360763     end:
%p A360763 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A360763 seq(T(n), n=0..12);
%t A360763 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 A360763 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 - 1, k], {k, 0, j}]]]];
%t A360763 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 A360763 T[n_] := CoefficientList[b[n, n], x];
%t A360763 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Sep 12 2023, after _Alois P. Heinz_ *)
%Y A360763 Columns k=0-2 give: A000007, A057427, A001477(n-1) for n>=1.
%Y A360763 Row sums give A089259.
%Y A360763 T(2n,n) gives A360784.
%Y A360763 T(3n,2n) gives A360785.
%Y A360763 Cf. A000009, A008289, A055884, A285229, A360742, A360764.
%K A360763 nonn,tabl
%O A360763 0,9
%A A360763 _Alois P. Heinz_, Feb 19 2023