OFFSET
0,7
COMMENTS
Rows and also columns reversed converge to A365441.
T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k < n or k > n*(n+1)/2.
LINKS
Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Partition of a set
FORMULA
EXAMPLE
T(4,7) = 5: 123|4, 124|3, 13|24, 14|23, 1|2|34.
T(5,9) = 10: 1234|5, 1235|4, 124|35, 125|34, 134|25, 135|24, 14|235, 15|234, 1|23|45, 1|245|3.
T(5,13) = 3: 1|23|4|5, 1|24|3|5, 1|25|3|4.
T(5,14) = 4: 12|3|4|5, 13|2|4|5, 14|2|3|5, 15|2|3|4.
T(5,15) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
1;
. 1;
. . 1, 1;
. . . 1, 1, 2, 1;
. . . . 1, 1, 2, 5, 2, 3, 1;
. . . . . 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1;
. . . . . . 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1;
...
MAPLE
b:= proc(n, m) option remember; `if`(n=0, 1,
b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
end:
T:= (n, k)-> coeff(b(n, 0), x, k):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, t^i, `if`(t=0, 0, t*b(n, i-1, t))+
(t+1)^max(0, 2*i-n-1)*b(n-i, min(n-i, i-1), t+1)))
end:
T:= (n, k)-> b(k, n, 0):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
T[0, 0] = 1; T[n_, k_] := b[k, n, 0];
Table[Table[T[n, k], {k, n, n*(n + 1)/2}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz's second Maple program *)
CROSSREFS
KEYWORD
nonn,tabf,changed
AUTHOR
Alois P. Heinz, Dec 05 2023
STATUS
approved