A362787 - OEIS

%I #8 May 05 2023 07:47:00

%S 1,1,0,1,2,0,1,6,24,0,1,12,120,720,0,1,20,360,5040,40320,0,1,30,840,

%T 20160,362880,3628800,0,1,42,1680,60480,1814400,39916800,479001600,0,

%U 1,56,3024,151200,6652800,239500800,6227020800,87178291200,0,1,72,5040,332640,19958400,1037836800,43589145600,1307674368000,20922789888000,0

%N Triangle read by rows, T(n, k) = (-1)^k * RisingFactorial(n, k) * FallingFactorial(k - n, k).

%F T(n, k) = Pochhammer(n, k) * Pochhammer(n - k, k).

%F T(n, k) = Gamma(n + k) / Gamma(n - k) if k != n.

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

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

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] 1,  0;

%e [2] 1,  2,    0;

%e [3] 1,  6,   24,      0;

%e [4] 1, 12,  120,    720,       0;

%e [5] 1, 20,  360,   5040,   40320,         0;

%e [6] 1, 30,  840,  20160,  362880,   3628800,          0;

%e [7] 1, 42, 1680,  60480, 1814400,  39916800,  479001600,           0;

%e [8] 1, 56, 3024, 151200, 6652800, 239500800, 6227020800, 87178291200, 0;

%p T := (n, k) -> if n = k then 0^k else GAMMA(n + k) / GAMMA(n - k) fi:

%p seq(seq(T(n, k), k = 0..n), n = 0..9);

%t Table[Pochhammer[n, k]*Pochhammer[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, May 05 2023 *)

%Y Cf. A362846 (row sums), A010050 (main diagonal), A002378 (column 1).

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, May 05 2023