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Square array T(n, k) = (k+2)^binomial(n, 2) with T(n, 0) = n!, read by antidiagonals.
1

%I #9 Sep 08 2022 08:45:41

%S 1,1,1,1,1,2,1,1,3,6,1,1,4,27,24,1,1,5,64,729,120,1,1,6,125,4096,

%T 59049,720,1,1,7,216,15625,1048576,14348907,5040,1,1,8,343,46656,

%U 9765625,1073741824,10460353203,40320,1,1,9,512,117649,60466176,30517578125,4398046511104,22876792454961,362880

%N Square array T(n, k) = (k+2)^binomial(n, 2) with T(n, 0) = n!, read by antidiagonals.

%H G. C. Greubel, <a href="/A156582/b156582.txt">Antidiagonal rows n = 0..50, flattened</a>

%F T(n,k) = Product_{j=1..n} ( Sum_{i=0..j-1} binomial(j-1, i)*(k+1)^i ) with T(n, 0) = n! (square array).

%F T(n, k) = (k+2)^binomial(n, 2) with T(n, 0) = n! (square array). - _G. C. Greubel_, Jun 28 2021

%e Square array begins as:

%e 1, 1, 1, 1, 1, 1 ...;

%e 1, 1, 1, 1, 1, 1 ...;

%e 2, 3, 4, 5, 6, 7 ...;

%e 6, 27, 64, 125, 216, 343 ...;

%e 24, 729, 4096, 15625, 46656, 117649 ...;

%e 120, 59049, 1048576, 9765625, 60466176, 282475249 ...;

%e Antidiagonal triangle begins as:

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 3, 6;

%e 1, 1, 4, 27, 24;

%e 1, 1, 5, 64, 729, 120;

%e 1, 1, 6, 125, 4096, 59049, 720;

%e 1, 1, 7, 216, 15625, 1048576, 14348907, 5040;

%e 1, 1, 8, 343, 46656, 9765625, 1073741824, 10460353203, 40320;

%t (* First program *)

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

%t Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 28 2021 *)

%t (* Second program *)

%t T[n_, k_]:= If[k==0, n!, (k+2)^Binomial[n, 2]];

%t Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 28 2021 *)

%o (Magma)

%o A156582:= func< n,k | k eq 0 select Factorial(n) else (k+2)^Binomial(n,2) >;

%o [A156582(k,n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 28 2021

%o (Sage)

%o def A156582(n,k): return factorial(n) if (k==0) else (k+2)^binomial(n,2)

%o flatten([[A156582(k,n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 28 2021

%Y Cf. A118180, A118185, A118190.

%K nonn,tabl

%O 0,6

%A _Roger L. Bagula_, Feb 10 2009

%E Edited by _G. C. Greubel_, Jun 28 2021