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A156582 Square array T(n, k) = (k+2)^binomial(n, 2) with T(n, 0) = n!, read by antidiagonals.

%I A156582 #9 Sep 08 2022 08:45:41
%S A156582 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 A156582 59049,720,1,1,7,216,15625,1048576,14348907,5040,1,1,8,343,46656,
%U A156582 9765625,1073741824,10460353203,40320,1,1,9,512,117649,60466176,30517578125,4398046511104,22876792454961,362880
%N A156582 Square array T(n, k) = (k+2)^binomial(n, 2) with T(n, 0) = n!, read by antidiagonals.
%H A156582 G. C. Greubel, <a href="/A156582/b156582.txt">Antidiagonal rows n = 0..50, flattened</a>
%F A156582 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 A156582 T(n, k) = (k+2)^binomial(n, 2) with T(n, 0) = n! (square array). - _G. C. Greubel_, Jun 28 2021
%e A156582 Square array begins as:
%e A156582     1,     1,       1,       1,        1,         1 ...;
%e A156582     1,     1,       1,       1,        1,         1 ...;
%e A156582     2,     3,       4,       5,        6,         7 ...;
%e A156582     6,    27,      64,     125,      216,       343 ...;
%e A156582    24,   729,    4096,   15625,    46656,    117649 ...;
%e A156582   120, 59049, 1048576, 9765625, 60466176, 282475249 ...;
%e A156582 Antidiagonal triangle begins as:
%e A156582   1;
%e A156582   1, 1;
%e A156582   1, 1, 2;
%e A156582   1, 1, 3,   6;
%e A156582   1, 1, 4,  27,    24;
%e A156582   1, 1, 5,  64,   729,     120;
%e A156582   1, 1, 6, 125,  4096,   59049,        720;
%e A156582   1, 1, 7, 216, 15625, 1048576,   14348907,        5040;
%e A156582   1, 1, 8, 343, 46656, 9765625, 1073741824, 10460353203, 40320;
%t A156582 (* First program *)
%t A156582 T[n_, k_]:= If[k==0, n!, Product[Sum[Binomial[j-1,i]*(k+1)^i, {i,0,j-1}], {j,n}]];
%t A156582 Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 28 2021 *)
%t A156582 (* Second program *)
%t A156582 T[n_, k_]:= If[k==0, n!, (k+2)^Binomial[n, 2]];
%t A156582 Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 28 2021 *)
%o A156582 (Magma)
%o A156582 A156582:= func< n,k | k eq 0 select Factorial(n) else (k+2)^Binomial(n,2) >;
%o A156582 [A156582(k,n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 28 2021
%o A156582 (Sage)
%o A156582 def A156582(n,k): return factorial(n) if (k==0) else (k+2)^binomial(n,2)
%o A156582 flatten([[A156582(k,n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 28 2021
%Y A156582 Cf. A118180, A118185, A118190.
%K A156582 nonn,tabl
%O A156582 0,6
%A A156582 _Roger L. Bagula_, Feb 10 2009
%E A156582 Edited by _G. C. Greubel_, Jun 28 2021