A354977 Triangle read by rows. T(n, k) = Sum_{j=0..n}((-1)^(n-j)*binomial(n, j)*j^(n+k)) / n!.

1, 1, 1, 1, 3, 7, 1, 6, 25, 90, 1, 10, 65, 350, 1701, 1, 15, 140, 1050, 6951, 42525, 1, 21, 266, 2646, 22827, 179487, 1323652, 1, 28, 462, 5880, 63987, 627396, 5715424, 49329280, 1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053

Offset: 0

Author

Peter Luschny, Jun 15 2022

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) begins:
[0] 1;
[1] 1,  1;
[2] 1,  3,   7;
[3] 1,  6,  25,    90;
[4] 1, 10,  65,   350,   1701;
[5] 1, 15, 140,  1050,   6951,   42525;
[6] 1, 21, 266,  2646,  22827,  179487,  1323652;
[7] 1, 28, 462,  5880,  63987,  627396,  5715424,  49329280;
[8] 1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053;
		

Links

• Andrew Francis and Michael Hendriksen, Counting spinal phylogenetic networks, arXiv:2502.14223 [q-bio.PE], 2025. See p. 13.

Crossrefs

T(n,1) = A000217, T(n,n) = A007820, A354978 (row sums), A048993.

Programs

Maple

T := (n, k) -> add((-1)^(n - j)*binomial(n, j)*j^(n + k), j = 0..n) / n!:
seq(seq(T(n, k), k = 0..n), n = 0..8);

Formula

T(n, k) = Stirling2(n + k, n).