OFFSET
0,2
COMMENTS
T(n,k) is defined for all 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 = 0..140, flattened
Wikipedia, Partition of a set
FORMULA
T(n,k) = Sum_{j=0..n-k} binomial(n,j) * Bell(j).
T(n,k) = Bell(n+1) - Sum_{j=0..k-1} binomial(n,j) * Bell(n-j).
Sum_{k=0..n} T(n,k) = n*Bell(n)+Bell(n+1) = A124427(n+1).
Sum_{k=1..n} T(n,k) = n*Bell(n) = A070071(n).
T(n,0)-T(n,1) = Bell(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A224271(n+1). - Alois P. Heinz, May 17 2023
EXAMPLE
T(3,2) = 4 because the number of blocks of size >= 2 in all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+1+1+1+0 = 4.
Triangle T(n,k) begins:
1;
2, 1;
5, 3, 1;
15, 10, 4, 1;
52, 37, 17, 5, 1;
203, 151, 76, 26, 6, 1;
877, 674, 362, 137, 37, 7, 1;
4140, 3263, 1842, 750, 225, 50, 8, 1;
21147, 17007, 9991, 4307, 1395, 345, 65, 9, 1;
...
MAPLE
T:= proc(n, k) option remember; `if`(k>n, 0,
binomial(n, k)*combinat[bell](n-k)+T(n, k+1))
end:
seq(seq(T(n, k), k=0..n), n=0..14);
MATHEMATICA
T[n_, k_] := Sum[Binomial[n, j]*BellB[j], {j, 0, n - k}];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2018 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 14 2017
STATUS
approved