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Triangle read by rows: T(n,k) is the number of n X k Young tableaux, where 0 <= k <= n.
2

%I #31 May 07 2021 00:51:41

%S 1,1,1,1,1,2,1,1,5,42,1,1,14,462,24024,1,1,42,6006,1662804,701149020,

%T 1,1,132,87516,140229804,396499770810,1671643033734960,1,1,429,

%U 1385670,13672405890,278607172289160,9490348077234178440,475073684264389879228560

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

%H Seiichi Manyama, <a href="/A321716/b321716.txt">Rows n = 0..30, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hook_length_formula">Hook length formula</a>

%H <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>

%F T(n, k) = (n*k)! / (Product_{i=1..n} Product_{j=1..k} (i+j-1)).

%F T(n, k) = A060854(n,k) for n,k > 0.

%F T(n, n) = A039622(n).

%F 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 T(4,3) = 12! / ((6*5*4)*(5*4*3)*(4*3*2)*(3*2*1)) = 462.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 5, 42;

%e 1, 1, 14, 462, 24024;

%e 1, 1, 42, 6006, 1662804, 701149020;

%e 1, 1, 132, 87516, 140229804, 396499770810, 1671643033734960;

%t 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 T[n_, k_]:= (n*k)!*BarnesG[n+1]*BarnesG[k+1]/BarnesG[n+k+1];

%t Table[T[n, k], {n, 0, 5}, {k, 0, n}] //Flatten (* _G. C. Greubel_, May 04 2021 *)

%o (Magma)

%o 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(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 04 2021

%o (Sage)

%o def A321716(n,k): return factorial(n*k)/product( gamma(j+k)/gamma(j) for j in (1..n) )

%o flatten([[A321716(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 04 2021

%Y Cf. A000178, A005789, A005790, A005791, A039622, A060854

%K nonn,tabl

%O 0,6

%A _Seiichi Manyama_, Nov 17 2018