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

%I A354796 #10 Jun 11 2022 06:33:18
%S A354796 1,1,1,1,2,3,2,6,12,20,6,24,60,120,210,24,120,360,840,1680,3024,120,
%T A354796 720,2520,6720,15120,30240,55440,720,5040,20160,60480,151200,332640,
%U A354796 665280,1235520,5040,40320,181440,604800,1663200,3991680,8648640,17297280,32432400
%N A354796 Triangle read by rows. T(n, k) = Gamma(k + n) / k! for n >= 1 and 0 <= k <= n, T(0, 0) = 1.
%F A354796 T(n, k) = binomial(n + k - 1, n - 1)*(n - 1)! for n >= 1.
%F A354796 T(n, n) = Sum_{k=0..n-1} T(n, k). Row sums are 2*A006963(n + 1) for n >= 1.
%e A354796 Table T(n, k) begins:
%e A354796 [0]   1;
%e A354796 [1]   1,    1;
%e A354796 [2]   1,    2,     3;
%e A354796 [3]   2,    6,    12,    20;
%e A354796 [4]   6,   24,    60,   120,    210;
%e A354796 [5]  24,  120,   360,   840,   1680,   3024;
%e A354796 [6] 120,  720,  2520,  6720,  15120,  30240,  55440;
%e A354796 [7] 720, 5040, 20160, 60480, 151200, 332640, 665280, 1235520;
%p A354796 T := (n, k) -> ifelse(n = 0, 1, GAMMA(k + n) / GAMMA(k + 1));
%p A354796 for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
%t A354796 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 A354796 Cf. A006963 (main diagonal),
%K A354796 nonn,tabl
%O A354796 0,5
%A A354796 _Peter Luschny_, Jun 11 2022