A156882 Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^3 - (k+1))^i ) with T(n, 0) = n!, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 7, 6, 1, 1, 25, 301, 24, 1, 1, 61, 15025, 77959, 120, 1, 1, 121, 223321, 216735625, 121226245, 720, 1, 1, 211, 1757041, 49054914181, 75034090110625, 1131162092095, 5040, 1, 1, 337, 9349621, 3061680840361, 646527139289672641, 623445123763413765625, 63330372050122765, 40320

Offset

0,6

Links

G. C. Greubel, Antidiagonal rows n = 0..25, flattened

Formula

T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^3 - (k+1))^i ) with T(n, 0) = n! (square array).

T(n, k) = ( Product_{j=1..n} ((k*(k+1)*(k+2))^j -1) )/(k*(k+1)*(k+2) -1)^n with T(n, 0) = n! (square array). - G. C. Greubel, Jun 14 2021

Example

Square array begins as:
   1,     1,         1,           1,             1,              1 ...;
   1,     1,         1,           1,             1,              1 ...;
   2,     7,        25,          61,           121,            211 ...;
   6,   301,     15025,      223321,       1757041,        9349621 ...;
  24, 77959, 216735625, 49054914181, 3061680840361, 87001131137131 ...;
Triangle begins as:
  1;
  1, 1;
  1, 1,   2;
  1, 1,   7,       6;
  1, 1,  25,     301,          24;
  1, 1,  61,   15025,       77959,            120;
  1, 1, 121,  223321,   216735625,      121226245,           720;
  1, 1, 211, 1757041, 49054914181, 75034090110625, 1131162092095, 5040;

Mathematica

(* First program *)
T[n_, m_] = If[m==0, n!, Product[Sum[(-(m+1) + (m+1)^3)^i, {i, 0, k-1}], {k, n}]];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 12 2021 *)
(* Second program *)
T[n_, k_]= If[k==0, n!, Product[((k*(k+1)*(k+2))^j -1), {j, n}]/(k*(k+1)*(k+2) - 1)^n];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 14 2021 *)

Prog

(Sage)
def A156882(n, k): return factorial(n) if (k==0) else product(((k*(k+1)*(k+2))^j -1) for j in (1..n))/(k*(k+1)*(k+2)-1)^n
flatten([[A156882(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2021

Crossrefs

Cf. A156881, A156883, A156885, A156888, A156889.

Sequence in context: A196832 A005450 A039760 * A266309 A122021 A303817

Adjacent sequences: A156879 A156880 A156881 * A156883 A156884 A156885

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 17 2009

Extensions

Edited by G. C. Greubel, Jun 14 2021

Status

approved