A156576 Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} (i+1)*(k+1)^i ) with T(n, 0) = n!, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 5, 6, 1, 1, 7, 85, 24, 1, 1, 9, 238, 4165, 120, 1, 1, 11, 513, 33796, 537285, 720, 1, 1, 13, 946, 160569, 18486412, 172468485, 5040, 1, 1, 15, 1573, 554356, 255786417, 37065256060, 132628264965, 40320, 1, 1, 17, 2430, 1549405, 2057215116, 1979019508329, 263459840074480, 237802479082245, 362880

Offset

0,6

Links

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Formula

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

T(n, k) = (1/k^(2*n))*Product_{j=1..n} (1 -(j+1)*(k+1)^j +j*(k+1)^(j+1)) with T(n, 0) = n! (square array). - G. C. Greubel, Jun 28 2021

Example

Square array begins as:
    1,      1,        1,         1,          1,           1 ...;
    1,      1,        1,         1,          1,           1 ...;
    2,      5,        7,         9,         11,          13 ...;
    6,     85,      238,       513,        946,        1573 ...;
   24,   4165,    33796,    160569,     554356,     1549405 ...;
  120, 537285, 18486412, 255786417, 2057215116, 11566308325 ...;
Antidiagonal triangle begins as:
  1;
  1, 1;
  1, 1,  2;
  1, 1,  5,    6;
  1, 1,  7,   85,     24;
  1, 1,  9,  238,   4165,       120;
  1, 1, 11,  513,  33796,    537285,         720;
  1, 1, 13,  946, 160569,  18486412,   172468485,         5040;
  1, 1, 15, 1573, 554356, 255786417, 37065256060, 132628264965, 40320;

Mathematica

(* First program *)
T[n_, k_]:= T[n, k]= If[k==0, n!, Product[Sum[(i+1)*(k+1)^i, {i, 0, j-1}] {j, n}]];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 28 2021 *)
(* Second program *)
T[n_, k_]:= If[k==0, n!, Product[1 -(j+1)*(k+1)^j +j*(k+1)^(j+1), {j, n}]/k^(2*n)];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 28 2021 *)

Prog

(Magma)
A156576:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (1/k^(2*n))*(&*[1 -(j+1)*(k+1)^j +j*(k+1)^(j+1): j in [1..n]]) >;
[A156576(k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021
(Sage)
def A156576(n, k): return factorial(n) if (k==0) else (1/k^(2*n))*product( 1 -(j+1)*(k+1)^j +j*(k+1)^(j+1) for j in [1..n])
flatten([[A156576(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

Crossrefs

Sequence in context: A086856 A052916 A326048 * A293219 A266572 A266681

Adjacent sequences: A156573 A156574 A156575 * A156577 A156578 A156579

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 10 2009

Extensions

Edited by G. C. Greubel, Jun 28 2021

Status

approved