OFFSET
0,8
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set
FORMULA
Sum_{k=0..n} k * T(n,k) = A372650(n).
EXAMPLE
T(5,1) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
T(5,2) = 10: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
T(5,3) = 10: 12|3|4|5, 13|2|4|5, 1|23|4|5, 14|2|3|5, 1|24|3|5, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
T(5,4) = 0.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 4, 0, 1;
0, 5, 9, 0, 1;
0, 31, 10, 10, 0, 1;
0, 82, 70, 35, 15, 0, 1;
0, 344, 336, 140, 35, 21, 0, 1;
0, 1661, 1393, 616, 385, 56, 28, 0, 1;
0, 7942, 6210, 4984, 1386, 504, 84, 36, 0, 1;
0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1;
...
MAPLE
b:= proc(n, m, t) option remember; `if`(n=0, x^t,
add(binomial(n-1, j-1)*b(n-j, min(j, m),
`if`(j<m, 1, `if`(j=m, t+1, t))), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 12 2024
STATUS
approved