OFFSET
1,8
COMMENTS
T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k < n. T(n,k) = 0 for k >= n.
LINKS
Alois P. Heinz, Rows n = 1..200, flattened
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 2, 1, 1;
0, 3, 4, 1, 1;
0, 6, 9, 3, 1, 1;
0, 12, 22, 9, 3, 1, 1;
0, 25, 54, 23, 8, 3, 1, 1;
0, 51, 139, 60, 23, 8, 3, 1, 1;
0, 111, 346, 166, 61, 22, 8, 3, 1, 1;
MAPLE
g:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
binomial(g(i-1$2, k)+j-1, j)*g(n-i*j, i-1, k), j=0..min(k, n/i))))
end:
T:= (n, k)-> g(n-1$2, k) -`if`(k=0, 0, g(n-1$2, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..14);
MATHEMATICA
g[n_, i_, k_] := g[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i - 1, i - 1, k] + j - 1, j]*g[n - i*j, i - 1, k], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := g[n - 1, n - 1, k] - If[k == 0, 0, g[n - 1, n - 1, k - 1]];
Table[T[n, k], {n, 1, 14}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 02 2018
STATUS
approved