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
Examples
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;
Programs
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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;
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.