A174159 Triangle read by rows. T(n, k) = 2 * Eulerian(n, k - 1) - binomial(n - 1, k - 1)* binomial(n, k - 1) / k.

1, 1, 1, 1, 5, 1, 1, 16, 16, 1, 1, 42, 112, 42, 1, 1, 99, 554, 554, 99, 1, 1, 219, 2277, 4657, 2277, 219, 1, 1, 466, 8390, 30748, 30748, 8390, 466, 1, 1, 968, 28880, 175292, 310616, 175292, 28880, 968, 1, 1, 1981, 95140, 907864, 2615416, 2615416, 907864, 95140

Offset: 1

Author

Roger L. Bagula, Mar 10 2010

Keywords

• nonn
• tabl

Examples

			[ 1] 1;
[ 2] 1,    1;
[ 3] 1,    5,     1;
[ 4] 1,   16,    16,      1;
[ 5] 1,   42,   112,     42,       1;
[ 6] 1,   99,   554,    554,      99,       1;
[ 7] 1,  219,  2277,   4657,    2277,     219,      1;
[ 8] 1,  466,  8390,  30748,   30748,    8390,    466,     1;
[ 9] 1,  968, 28880, 175292,  310616,  175292,  28880,   968,    1;
[10] 1, 1981, 95140, 907864, 2615416, 2615416, 907864, 95140, 1981, 1;
		

Crossrefs

Cf. A001263, A008292, A356118 (row sums).

Programs

Maple

# Works also if based on (0, 0).
T := (n,k) -> `if`(k = 0, k^n, 2*combinat:-eulerian1(n, k-1) - binomial(n-1, k-1)* binomial(n, k-1) / k):
for n from 1 to 6 do seq(T(n,k), k=1..n) od;  # Peter Luschny, Jul 27 2022

Mathematica

Needs["Combinatorica`"];
T[n_, m_] := 2*Eulerian[n, m - 1] - Binomial[n - 1, m - 1]*Binomial[n, m - 1]/m;
Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten

Extensions

Edited by Peter Luschny, Jul 27 2022