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

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, 59049, 720, 1, 1, 7, 216, 15625, 1048576, 14348907, 5040, 1, 1, 8, 343, 46656, 9765625, 1073741824, 10460353203, 40320, 1, 1, 9, 512, 117649, 60466176, 30517578125, 4398046511104, 22876792454961, 362880

Offset: 0

Author

Roger L. Bagula, Feb 10 2009

Keywords

• nonn
• tabl

Examples

			Square array begins as:
    1,     1,       1,       1,        1,         1 ...;
    1,     1,       1,       1,        1,         1 ...;
    2,     3,       4,       5,        6,         7 ...;
    6,    27,      64,     125,      216,       343 ...;
   24,   729,    4096,   15625,    46656,    117649 ...;
  120, 59049, 1048576, 9765625, 60466176, 282475249 ...;
Antidiagonal triangle begins as:
  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,   59049,        720;
  1, 1, 7, 216, 15625, 1048576,   14348907,        5040;
  1, 1, 8, 343, 46656, 9765625, 1073741824, 10460353203, 40320;
		

Links

• G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Crossrefs

Cf. A118180, A118185, A118190.

Programs

Magma

A156582:= func< n,k | k eq 0 select Factorial(n) else (k+2)^Binomial(n,2) >;
[A156582(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021

Mathematica

(* First program *)
T[n_, k_]:= If[k==0, n!, Product[Sum[Binomial[j-1,i]*(k+1)^i, {i,0,j-1}], {j,n}]];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 28 2021 *)
(* Second program *)
T[n_, k_]:= If[k==0, n!, (k+2)^Binomial[n, 2]];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 28 2021 *)

Sage

def A156582(n,k): return factorial(n) if (k==0) else (k+2)^binomial(n,2)
flatten([[A156582(k,n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

Formula

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).

T(n, k) = (k+2)^binomial(n, 2) with T(n, 0) = n! (square array). - G. C. Greubel, Jun 28 2021

Extensions

Edited by G. C. Greubel, Jun 28 2021