A354796 Triangle read by rows. T(n, k) = Gamma(k + n) / k! for n >= 1 and 0 <= k <= n, T(0, 0) = 1.

1, 1, 1, 1, 2, 3, 2, 6, 12, 20, 6, 24, 60, 120, 210, 24, 120, 360, 840, 1680, 3024, 120, 720, 2520, 6720, 15120, 30240, 55440, 720, 5040, 20160, 60480, 151200, 332640, 665280, 1235520, 5040, 40320, 181440, 604800, 1663200, 3991680, 8648640, 17297280, 32432400

Offset: 0

Author

Peter Luschny, Jun 11 2022

Keywords

• nonn
• tabl

Examples

			Table T(n, k) begins:
[0]   1;
[1]   1,    1;
[2]   1,    2,     3;
[3]   2,    6,    12,    20;
[4]   6,   24,    60,   120,    210;
[5]  24,  120,   360,   840,   1680,   3024;
[6] 120,  720,  2520,  6720,  15120,  30240,  55440;
[7] 720, 5040, 20160, 60480, 151200, 332640, 665280, 1235520;
		

Crossrefs

Cf. A006963 (main diagonal),

Programs

Maple

T := (n, k) -> ifelse(n = 0, 1, GAMMA(k + n) / GAMMA(k + 1));
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

Mathematica

T[0, 0] = 1; T[n_, k_] := Gamma[n + k]/Gamma[k + 1]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 11 2022 *)

Formula

T(n, k) = binomial(n + k - 1, n - 1)*(n - 1)! for n >= 1.

T(n, n) = Sum_{k=0..n-1} T(n, k). Row sums are 2*A006963(n + 1) for n >= 1.