A156540 Array A(n, k) = Product_{j=1..n} ( (k+1)^j - 1 ) with A(n, 0) = n!, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 21, 24, 1, 1, 5, 52, 315, 120, 1, 1, 6, 105, 2080, 9765, 720, 1, 1, 7, 186, 8925, 251680, 615195, 5040, 1, 1, 8, 301, 29016, 3043425, 91611520, 78129765, 40320, 1, 1, 9, 456, 77959, 22661496, 4154275125, 100131391360, 19923090075, 362880

Offset

0,6

Links

G. C. Greubel, antidiagonals n = 0..50, flattened

Formula

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

Example

Triangle begins as:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 3,   6;
  1, 1, 4,  21,    24;
  1, 1, 5,  52,   315,      120;
  1, 1, 6, 105,  2080,     9765,        720;
  1, 1, 7, 186,  8925,   251680,     615195,         5040;
  1, 1, 8, 301, 29016,  3043425,   91611520,     78129765,       40320;
  1, 1, 9, 456, 77959, 22661496, 4154275125, 100131391360, 19923090075, 362880;

Mathematica

A[n_, k_]:= If[k==0, n!, (1/k^n)*Product[ (k + 1)^j -1, {j, n}] ];
T[n_, k_]:= A[k, n-k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//TableForm (* modified by G. C. Greubel, Jan 04 2022 *)

Prog

(Sage)
def A(n, k): return factorial(n) if (k==0) else (1/k^n)*product( (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. A156579.

Sequence in context: A174480 A111670 A123353 * A156582 A156953 A293785

Adjacent sequences: A156537 A156538 A156539 * A156541 A156542 A156543

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 09 2009

Extensions

Edited by G. C. Greubel, Jan 04 2022

Status

approved