OFFSET
0,9
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
T(n,n) + T(n+1,n) = T(n+2,n+1) for n>=0.
EXAMPLE
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;
...
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 18 2023
STATUS
approved