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

0,6

Links

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

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

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A118180, A118185, A118190.

Sequence in context: A111670 A123353 A156540 * A156953 A293785 A156881

Adjacent sequences: A156579 A156580 A156581 * A156583 A156584 A156585

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 10 2009

Extensions

Edited by G. C. Greubel, Jun 28 2021

Status

approved