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Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-k*i| <= n} (n-k*i), with n >= 0, k >= 1.
10

%I #18 Apr 05 2020 09:34:10

%S 1,1,-1,1,-1,4,1,1,-4,-36,1,1,-2,9,576,1,1,-4,-9,64,-14400,1,1,2,-3,

%T -8,-225,518400,1,1,2,-6,-16,40,-2304,-25401600,1,1,2,-9,-4,-15,324,

%U 11025,1625702400,1,1,2,3,-8,-25,144,280,147456,-131681894400,1,1,2,3,-12,-5,-24,105,-2240,-893025,13168189440000

%N Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-k*i| <= n} (n-k*i), with n >= 0, k >= 1.

%D F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.

%H J. Dezert, ed., <a href="http://www.mathematicsmagazine.com/corresp/J_Dezert/JDezert.htm">Smarandacheials (1)</a>, Mathematics Magazine for Grades 1-12, No. 4, 2004.

%H J. Dezert, ed., <a href="http://www.mathematicsmagazine.com/corresp/J_Dezert/JDezert2.htm">Smarandacheials (2)</a>, Mathematics Magazine for Grades 1-12, No. 4, 2004.

%e Array begins:

%e 1, -1, 4, -36, 576, -14400, 518400, -25401600, 1625702400, -131681894400, ...

%e 1, -1, -4, 9, 64, -225, -2304, 11025, 147456, -893025, -14745600, 108056025, ...

%e 1, 1, -2, -9, -8, 40, 324, 280, -2240, -26244, -22400, 246400, 3779136, ...

%e 1, 1, -4, -3, -16, -15, 144, 105, 1024, 945, -14400, -10395, -147456, ...

%e 1, 1, 2, -6, -4, -25, -24, -42, 336, 216, 2500, 2376, 4032, ...

%e 1, 1, 2, -9, -8, -5, -36, -35, -64, 729, 640, 385, 5184, ...

%e 1, 1, 2, 3, -12, -10, -6, -49, -48, -90, -120, 1320, 1080, ...

%e 1, 1, 2, 3, -16, -15, -12, -7, -64, -63, -120, -165, 2304, ...

%e 1, 1, 2, 3, 4, -20, -18, -14, -8, -81, -80, -154, -216, ...

%e 1, 1, 2, 3, 4, -25, -24, -21, -16, -9, -100, -99, -192, ...

%e ...

%p T:=proc(n,k) local i,p;

%p p:=1;

%p for i from 0 to floor(2*n/k) do

%p if n-k*i <> 0 then p:=p*(n-k*i) fi; od:

%p p;

%p end;

%p scan1:=proc(a,M1) local lis,n,k; lis:=[]; for n from 1 to M1 do for k from 0 to n-1 do

%p lis:=[op(lis),a(k,n-k)]; od: od: lis; end:

%p scan1(T,12);

%t T[n_, k_] := Module[{i, p = 1}, For[i = 0, i <= Floor[2n/k], i++, If[n - k i != 0, p *= (n - k i)]]; p]; T[_, 0] = 1;

%t Table[T[k, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 05 2020, after Maple *)

%Y Rows k=1 through 9 are signed A001044 or A092396, signed A184877 or A092397, A092398, A092399, A092971, A092972, A092973, A092974,

%K sign,tabl

%O 0,6

%A _N. J. A. Sloane_, Jul 03 2017