A156881 Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^2 - (k+1))^i ) with T(n, 0) = n!, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 7, 21, 24, 1, 1, 13, 301, 315, 120, 1, 1, 21, 2041, 77959, 9765, 720, 1, 1, 31, 8841, 3847285, 121226245, 615195, 5040, 1, 1, 43, 28861, 74450061, 87029433985, 1131162092095, 78129765, 40320, 1, 1, 57, 77701, 806116591, 12538953723681, 23624400943530205, 63330372050122765, 19923090075, 362880

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)^2 - (k+1))^i ) with T(n, 0) = n! (square array).

T(n, k) = ( Product_{j=1..n} (k^j*(k+1)^j -1) )/(k^2 + k -1)^n with T(n, 0) = n! (square array). - G. C. Greubel, Jun 12 2021

Example

Square array begins as:
   1,   1,     1,       1,        1, ...;
   1,   1,     1,       1,        1, ...;
   2,   3,     7,      13,       21, ...;
   6,  21,   301,    2041,     8841, ...;
  24, 315, 77959, 3847285, 74450061, ...;
Triangle begins as:
  1;
  1, 1;
  1, 1,  2;
  1, 1,  3,     6;
  1, 1,  7,    21,       24;
  1, 1, 13,   301,      315,         120;
  1, 1, 21,  2041,    77959,        9765,           720;
  1, 1, 31,  8841,  3847285,   121226245,        615195,     5040;
  1, 1, 43, 28861, 74450061, 87029433985, 1131162092095, 78129765, 40320;

Mathematica

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

Prog

(Sage)
def A156881(n, k): return factorial(n) if (k==0) else product((k^j*(k+1)^j -1) for j in (1..n))/(k^2 +k-1)^n
flatten([[A156881(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 12 2021

Crossrefs

Cf. A156882, A156883.

Sequence in context: A156582 A156953 A293785 * A056646 A056056 A136462

Adjacent sequences: A156878 A156879 A156880 * A156882 A156883 A156884

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 17 2009

Extensions

Edited by G. C. Greubel, Jun 12 2021

Status

approved