A156883 Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^4 - (k+1)^3)^i ) with T(n, 0) = n!, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 9, 6, 1, 1, 55, 657, 24, 1, 1, 193, 163405, 384345, 120, 1, 1, 501, 7152001, 26215881175, 1799118945, 720, 1, 1, 1081, 125501001, 50886093754945, 227121050616681925, 67375205371305, 5040, 1, 1, 2059, 1262046961, 15719063251251501, 69513937650491307135745, 106253703835242139200091375, 20185139902805378865, 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)^4 - (k+1)^3)^i ) with T(n, 0) = n! (square array).

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

Example

Square array begins as:
   1,      1,           1,              1, ...;
   1,      1,           1,              1, ...;
   2,      9,          55,            193, ...;
   6,    657,      163405,        7152001, ...;
  24, 384345, 26215881175, 50886093754945, ...;
Triangle begins as:
  1;
  1, 1;
  1, 1,    2;
  1, 1,    9,         6;
  1, 1,   55,       657,             24;
  1, 1,  193,    163405,         384345,                120;
  1, 1,  501,   7152001,    26215881175,         1799118945,            720;
  1, 1, 1081, 125501001, 50886093754945, 227121050616681925, 67375205371305, 5040;

Mathematica

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

Prog

(Sage)
def A156883(n, k): return factorial(n) if (k==0) else product((k^j*(k+1)^(3*j) -1) for j in (1..n))/(k*(k+1)^3 -1)^n
flatten([[A156883(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2021

Crossrefs

Cf. A156881, A156882, A156885, A156888, A156889.

Sequence in context: A124522 A016540 A132620 * A368924 A019803 A214506

Adjacent sequences: A156880 A156881 A156882 * A156884 A156885 A156886

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 17 2009

Extensions

Edited by G. C. Greubel, Jun 14 2021

Status

approved