A156889 Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^6 -(k+1)^5 -(k+1)^4 +(k+1)^2 )^i ) with T(n, 0) = n!, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 21, 6, 1, 1, 415, 8841, 24, 1, 1, 2833, 71301565, 74450061, 120, 1, 1, 11901, 22729320481, 5071662849566575, 12538953723681, 720, 1, 1, 37621, 1685442243801, 516439650916945061425, 149348900281032409928364325, 42236475040875277701, 5040

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)^6 -(k+1)^5 -(k+1)^4 +(k+1)^2 )^i ) with T(n, 0) = n! (square array).

T(n, k) = ( Product_{j=1..n} (f(k)^j -1) )/(f(k) -1)^n with T(n, 0) = n! and f(k) = k*(k+1)^2*(k^3 -3*k^2 -2*k -1) (square array). - G. C. Greubel, Jun 14 2021

Example

Square array begins as:
   1,        1,                1,                     1, ...;
   1,        1,                1,                     1, ...;
   2,       21,              415,                  2833, ...;
   6,     8841,         71301565,           22729320481, ...;
  24, 74450061, 5071662849566575, 516439650916945061425, ...;
Antidiagonal triangle begins as:
  1;
  1, 1;
  1, 1,     2;
  1, 1,    21,           6;
  1, 1,   415,        8841,               24;
  1, 1,  2833,    71301565,         74450061,            120;
  1, 1, 11901, 22729320481, 5071662849566575, 12538953723681, 720; ...

Mathematica

(* First program *)
T[n_, m_] = If[m==0, n!, Product[Sum[((m+1)^6 -(m+1)^5 -(m+1)^4 +(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 14 2021 *)
(* Second program *)
f[n_]:= n*(n+1)^2*(n^3 +3*n^2 +2*n -1);
T[n_, k_]= If[k==0, n!, Product[(f[k]^j -1), {j, n}]/(f[k]-1)^n];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 14 2021 *)

Prog

(Sage)
def f(n): return n*(n+1)^2*(n^3 +3*n^2 +2*n -1)
def A156889(n, k): return factorial(n) if (k==0) else product( (f(k)^j - 1) for j in (1..n))/( f(k) -1 )^n
flatten([[A156889(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2021

Crossrefs

Cf. A156881, A156882, A156883, A156885, A156888.

Sequence in context: A174966 A157453 A174174 * A172177 A156725 A141904

Adjacent sequences: A156886 A156887 A156888 * A156890 A156891 A156892

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 17 2009

Extensions

Edited by Joerg Arndt and G. C. Greubel, Jun 14 2021

Status

approved