A360205 Triangle read by rows. T(n, k) = (-1)^(n-k)*(k+1)*binomial(n, k)*pochhammer(1-n, n-k).

1, 0, 2, 0, 4, 3, 0, 12, 18, 4, 0, 48, 108, 48, 5, 0, 240, 720, 480, 100, 6, 0, 1440, 5400, 4800, 1500, 180, 7, 0, 10080, 45360, 50400, 21000, 3780, 294, 8, 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9, 0, 725760, 4354560, 6773760, 4233600, 1270080, 197568, 16128, 648, 10

Offset: 0

Author

Peter Luschny, Feb 08 2023

Keywords

• nonn
• tabl

Comments

A refinement of the number of partial permutations of an n-set (A002720).

Also the coefficients of a shifted derivative of the unsigned Lah polynomials (A271703).

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 0,     2;
[2] 0,     4,      3;
[3] 0,    12,     18,      4;
[4] 0,    48,    108,     48,      5;
[5] 0,   240,    720,    480,    100,     6;
[6] 0,  1440,   5400,   4800,   1500,   180,    7;
[7] 0, 10080,  45360,  50400,  21000,  3780,  294,   8;
[8] 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9;
		

Crossrefs

Cf. A052849 (column 1), A045991 (subdiagonal), A002720 (row sums), A271703.

Cf. A069138 (Stirling2 counterpart), A360174 (Stirling1 counterpart).

Programs

Maple

T := (n, k) -> (-1)^(n - k)*(k + 1)*binomial(n, k)*pochhammer(1 - n, n - k):
seq(seq(T(n, k), k = 0..n), n = 0..9);