A156885 Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^5 - (k+1)^4 - (k+1)^3 + (k+1)^2)^i ) with T(n, 0) = n!, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 13, 6, 1, 1, 145, 2041, 24, 1, 1, 721, 3027745, 3847285, 120, 1, 1, 2401, 374286241, 9104020469425, 87029433985, 720, 1, 1, 6301, 13835524801, 139895890728482161, 3941936722370875247425, 23624400943530205, 5040, 1, 1

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)^5 - (k+1)^4 - (k+1)^3 + (k+1)^2)^i ) with T(n, 0) = n! (square array).

T(n, k) = ( Product_{j=1..n} ((k^2*(k+1)^2*(k+2))^j -1) )/(k^2*(k+1)^2*(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, ...;
   2,      13,           145,                721, ...;
   6,    2041,       3027745,          374286241, ...;
  24, 3847285, 9104020469425, 139895890728482161, ...;
Antidiagonal triangle begins as:
  1;
  1, 1;
  1, 1,    2;
  1, 1,   13,         6;
  1, 1,  145,      2041,            24;
  1, 1,  721,   3027745,       3847285,         120;
  1, 1, 2401, 374286241, 9104020469425, 87029433985, 720; ...

Mathematica

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

Prog

(Sage)
def A156885(n, k): return factorial(n) if (k==0) else product(( (k^2*(k+1)^2*(k+2))^j -1) for j in (1..n))/(k^2*(k+1)^2*(k+2) -1)^n
flatten([[A156885(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2021

Crossrefs

Cf. A156881, A156882, A156883, A156888, A156889.

Sequence in context: A010246 A186430 A173889 * A174718 A176291 A337514

Adjacent sequences: A156882 A156883 A156884 * A156886 A156887 A156888

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 17 2009

Extensions

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

Status

approved