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