A156579 Array A(n, k) = Product_{j=1..n} ( j - (1+j)*(k+1) + (k+1)^(j+1) ) with A(n, 0) = n!, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 5, 44, 24, 1, 1, 6, 90, 1144, 120, 1, 1, 7, 162, 5220, 65208, 720, 1, 1, 8, 266, 18144, 934380, 7824960, 5040, 1, 1, 9, 408, 51604, 8219232, 507368340, 1932765120, 40320, 1, 1, 10, 594, 126480, 50313900, 14942563776, 830054604240, 970248090240, 362880

Offset

0,6

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = A(k, n-k) for the array defined by A(n, k) = Product_{j=1..n} ( j - (1+j)*(k+1) + (k+1)^(j+1) - 1 ) with A(n, 0) = n!.

Example

Triangle begins as:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 4,   6;
  1, 1, 5,  44,    24;
  1, 1, 6,  90,  1144,     120;
  1, 1, 7, 162,  5220,   65208,       720;
  1, 1, 8, 266, 18144,  934380,   7824960,       5040;
  1, 1, 9, 408, 51604, 8219232, 507368340, 1932765120, 40320;

Mathematica

A[n_, k_]:= If[k==0, n!, k^(-2*n)*Product[j -(1+j)*(k+1) +(k+1)^(j+1), {j, n}] ];
T[n_, k_]:= A[k, n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 04 2022 *)

Prog

(Sage)
def A(n, k): return factorial(n) if (k==0) else (1/k^(2*n))*product( j -(1+j)*(k+1) +(k+1)^(j+1) for j in (1..n) )
def T(n, k): return A(k, n-k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 04 2022

Crossrefs

Cf. A156540.

Sequence in context: A355334 A295259 A255009 * A351790 A322266 A190284

Adjacent sequences: A156576 A156577 A156578 * A156580 A156581 A156582

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 10 2009

Extensions

Edited by G. C. Greubel, Jan 04 2022

Status

approved