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Triangle T(n, k) = Product_{j=1..k} Product_{i=0..j-1} ( 1 - (n-k+1)*(2*i-1) ) with T(n, 0) = 1 and T(n, n) = n!, read by rows.
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%I #10 Feb 25 2021 08:31:53

%S 1,1,1,1,3,2,1,4,-9,6,1,5,-32,-135,24,1,6,-75,-2048,18225,120,1,7,

%T -144,-12375,1835008,31984875,720,1,8,-245,-48384,38795625,

%U 32883343360,-954268745625,5040,1,9,-384,-145775,390168576,3283855678125,-15321007338291200,-597882768540159375,40320

%N Triangle T(n, k) = Product_{j=1..k} Product_{i=0..j-1} ( 1 - (n-k+1)*(2*i-1) ) with T(n, 0) = 1 and T(n, n) = n!, read by rows.

%C Row sums are: 1, 2, 6, 2, -137, 16229, 33808092, -921346650220, -613200491632709703, 9136424641471148255125435, ...

%H G. C. Greubel, <a href="/A156699/b156699.txt">Rows n = 0..30 of the triangle, flattened</a>

%F Let the square array t(n, k) be given by t(n, k) = Product_{j=1..n} Product_{i=0..j-1} ( 1 - (k+1)*(2*i -1) ) with t(n, 0) = n!. The number triangle, T(n, k), is the downward antidiagonals, i.e. T(n, k) = t(k, n-k).

%F T(n, k) = (-2*(n-k+1))^binomial(k, 2)*(n-k+2)^k*Product_{j=1..k} Pochhammer( (n-k)/(2*(n-k+1)), j-1) with T(n, 0) = 1 and T(n, n) = n!. - _G. C. Greubel_, Feb 25 2021

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 3, 2;

%e 1, 4, -9, 6;

%e 1, 5, -32, -135, 24;

%e 1, 6, -75, -2048, 18225, 120;

%e 1, 7, -144, -12375, 1835008, 31984875, 720;

%e 1, 8, -245, -48384, 38795625, 32883343360, -954268745625, 5040;

%t (* First program *)

%t t[n_, k_]:= If[k==0, n!, Product[1 -(2*i-1)*(k+1), {j, n}, {i,0,j-1}]];

%t Table[t[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Feb 25 2021 *)

%t (* Second program *)

%t T[n_, k_] = If[k==0, 1, If[k==n, n!, (-2*(n-k+1))^Binomial[k, 2]*(n-k+2)^k *Product[Pochhammer[(n-k)/(2*(n-k+1)), j-1], {j,k}] ]];

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Feb 25 2021 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if (k==0): return 1

%o elif (k==n): return factorial(n)

%o else: return (-2*(n-k+1))^binomial(k, 2)*(n-k+2)^k*product( rising_factorial( (n-k)/(2*(n-k+1)), j-1) for j in (1..k))

%o flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 24 2021

%o (Magma)

%o function T(n,k)

%o if k eq 0 then return 1;

%o elif k eq n then return Factorial(n);

%o else return (&*[ (&*[1 - (n-k+1)*(2*m-1): m in [0..j-1]]) :j in [1..k]]);

%o end if; return T;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 25 2021

%Y Cf. A156698, A156730.

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 13 2009

%E Edited by _G. C. Greubel_, Feb 25 2021