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A321716 Triangle read by rows: T(n,k) is the number of n X k Young tableaux, where 0 <= k <= n.

%I A321716 #31 May 07 2021 00:51:41
%S A321716 1,1,1,1,1,2,1,1,5,42,1,1,14,462,24024,1,1,42,6006,1662804,701149020,
%T A321716 1,1,132,87516,140229804,396499770810,1671643033734960,1,1,429,
%U A321716 1385670,13672405890,278607172289160,9490348077234178440,475073684264389879228560
%N A321716 Triangle read by rows: T(n,k) is the number of n X k Young tableaux, where 0 <= k <= n.
%H A321716 Seiichi Manyama, <a href="/A321716/b321716.txt">Rows n = 0..30, flattened</a>
%H A321716 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hook_length_formula">Hook length formula</a>
%H A321716 <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>
%F A321716 T(n, k) = (n*k)! / (Product_{i=1..n} Product_{j=1..k} (i+j-1)).
%F A321716 T(n, k) = A060854(n,k) for n,k > 0.
%F A321716 T(n, n) = A039622(n).
%F A321716 T(n, k) = (n*k)!*BarnesG(n+1)*BarnesG(k+1)/BarnesG(n+k+1), where BarnesG(n) = A000178. - _G. C. Greubel_, May 04 2021
%e A321716 T(4,3) = 12! / ((6*5*4)*(5*4*3)*(4*3*2)*(3*2*1)) = 462.
%e A321716 Triangle begins:
%e A321716   1;
%e A321716   1, 1;
%e A321716   1, 1,   2;
%e A321716   1, 1,   5,    42;
%e A321716   1, 1,  14,   462,     24024;
%e A321716   1, 1,  42,  6006,   1662804,    701149020;
%e A321716   1, 1, 132, 87516, 140229804, 396499770810, 1671643033734960;
%t A321716 T[n_, k_]:= (n*k)!/Product[Product[i+j-1, {j,1,k}], {i,1,n}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Nov 17 2018 *)
%t A321716 T[n_, k_]:= (n*k)!*BarnesG[n+1]*BarnesG[k+1]/BarnesG[n+k+1];
%t A321716 Table[T[n, k], {n, 0, 5}, {k, 0, n}] //Flatten (* _G. C. Greubel_, May 04 2021 *)
%o A321716 (Magma)
%o A321716 A321716:= func< n,k | n eq 0 select 1 else Factorial(n*k)/(&*[ Round(Gamma(j+k)/Gamma(j)): j in [1..n]]) >;
%o A321716 [A321716(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 04 2021
%o A321716 (Sage)
%o A321716 def A321716(n,k): return factorial(n*k)/product( gamma(j+k)/gamma(j) for j in (1..n) )
%o A321716 flatten([[A321716(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 04 2021
%Y A321716 Cf. A000178, A005789, A005790, A005791, A039622, A060854
%K A321716 nonn,tabl
%O A321716 0,6
%A A321716 _Seiichi Manyama_, Nov 17 2018