%I #10 Jun 11 2022 06:33:18
%S 1,1,1,1,2,3,2,6,12,20,6,24,60,120,210,24,120,360,840,1680,3024,120,
%T 720,2520,6720,15120,30240,55440,720,5040,20160,60480,151200,332640,
%U 665280,1235520,5040,40320,181440,604800,1663200,3991680,8648640,17297280,32432400
%N Triangle read by rows. T(n, k) = Gamma(k + n) / k! for n >= 1 and 0 <= k <= n, T(0, 0) = 1.
%F T(n, k) = binomial(n + k - 1, n - 1)*(n - 1)! for n >= 1.
%F T(n, n) = Sum_{k=0..n-1} T(n, k). Row sums are 2*A006963(n + 1) for n >= 1.
%e Table T(n, k) begins:
%e [0] 1;
%e [1] 1, 1;
%e [2] 1, 2, 3;
%e [3] 2, 6, 12, 20;
%e [4] 6, 24, 60, 120, 210;
%e [5] 24, 120, 360, 840, 1680, 3024;
%e [6] 120, 720, 2520, 6720, 15120, 30240, 55440;
%e [7] 720, 5040, 20160, 60480, 151200, 332640, 665280, 1235520;
%p T := (n, k) -> ifelse(n = 0, 1, GAMMA(k + n) / GAMMA(k + 1));
%p for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
%t 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 *)
%Y Cf. A006963 (main diagonal),
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Jun 11 2022