A362588 Triangle read by rows, T(n, k) = RisingFactorial(n - k, k) * FallingFactorial(n, k).

1, 1, 0, 1, 2, 0, 1, 6, 12, 0, 1, 12, 72, 144, 0, 1, 20, 240, 1440, 2880, 0, 1, 30, 600, 7200, 43200, 86400, 0, 1, 42, 1260, 25200, 302400, 1814400, 3628800, 0, 1, 56, 2352, 70560, 1411200, 16934400, 101606400, 203212800, 0

Offset

0,5

Links

Table of n, a(n) for n=0..44.

Formula

T(n, k) = (-1)^k * Pochhammer(n - k, k) * Pochhammer(-n, k).

T(n, k) = binomial(n, k) * binomial(n - 1, k) * (k!)^2.

Example

Table T(n, k) begins:
[0] 1;
[1] 1,  0;
[2] 1,  2,    0;
[3] 1,  6,   12,     0;
[4] 1, 12,   72,   144,       0;
[5] 1, 20,  240,  1440,    2880,        0;
[6] 1, 30,  600,  7200,   43200,    86400,         0;
[7] 1, 42, 1260, 25200,  302400,  1814400,   3628800,         0;
[8] 1, 56, 2352, 70560, 1411200, 16934400, 101606400, 203212800, 0;

Maple

T := (n, k) -> (-1)^k*pochhammer(n - k, k)*pochhammer(-n, k):
for n from 0 to 6 do seq(T(n, k), k=0..n) od;

Crossrefs

Cf. A228229 (row sums), A002378 (column 1), A010790 (subdiagonal).

Sequence in context: A226573 A260693 A337107 * A367073 A176129 A362787

Adjacent sequences: A362585 A362586 A362587 * A362589 A362590 A362591

Keyword

nonn,tabl

Author

Peter Luschny, May 05 2023

Status

approved