A343804 T(n, k) = Sum_{j=k..n} binomial(n, j)*E2(j, j-k), where E2 are the Eulerian numbers A201637. Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 1, 1, 1, 4, 1, 1, 15, 11, 1, 1, 64, 96, 26, 1, 1, 325, 824, 448, 57, 1, 1, 1956, 7417, 6718, 1779, 120, 1, 1, 13699, 71595, 96633, 43411, 6429, 247, 1, 1, 109600, 746232, 1393588, 944618, 243928, 21898, 502, 1, 1, 986409, 8403000, 20600856, 19521210, 7739362, 1250774, 71742, 1013, 1

Offset: 0

Author

Peter Luschny, Apr 30 2021

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1
[1] 1, 1
[2] 1, 4,      1
[3] 1, 15,     11,      1
[4] 1, 64,     96,      26,       1
[5] 1, 325,    824,     448,      57,       1
[6] 1, 1956,   7417,    6718,     1779,     120,     1
[7] 1, 13699,  71595,   96633,    43411,    6429,    247,     1
[8] 1, 109600, 746232,  1393588,  944618,   243928,  21898,   502,   1
[9] 1, 986409, 8403000, 20600856, 19521210, 7739362, 1250774, 71742, 1013, 1
		

Crossrefs

Row sums: A084262.

Cf. A046802 (Eulerian first order).

Programs

Maple

T := (n, k) -> add(binomial(n, r)*combinat:-eulerian2(r, r-k), r = k..n):
seq(seq(T(n, k), k = 0..n), n = 0..9);