A360742 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.
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, 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, 67, 94, 97, 79, 50, 25, 10, 0, 1, 10, 40, 93, 147, 172, 157, 117, 69, 33, 12
Offset: 0
Examples
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]}.
Triangle T(n,k) begins:
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, 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, 67, 94, 97, 79, 50, 25, 10;
...
Links
- Alois P. Heinz, Rows n = 0..200, flattened
Crossrefs
Programs
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Maple
h:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i))))) end: g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add( g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i), k), k=0..j)))) end: b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i)))) end: T:= (n, k)-> coeff(b(n$2), x, k): seq(seq(T(n, k), k=0..n), n=0..12); -
Mathematica
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]]]]]; 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}]]]]; 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[n_, k_] := Coefficient[b[n, n], x, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 15 2023, after Alois P. Heinz *)
Formula
T(n,n) + T(n+1,n) = T(n+2,n+1) for n>=0.