A090441 Symmetric triangle of certain normalized products of decreasing factorials.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 12, 6, 1, 1, 24, 144, 144, 24, 1, 1, 120, 2880, 8640, 2880, 120, 1, 1, 720, 86400, 1036800, 1036800, 86400, 720, 1, 1, 5040, 3628800, 217728000, 870912000, 217728000, 3628800, 5040, 1, 1, 40320, 203212800, 73156608000

Offset

-1,8

Comments

Similar to, but different from, superfactorial Pascal triangle A009963.

A009963(n,m) = (Product_{p=0..m-1} (n-p)!)/superfac(m) with n >= m >= 0, otherwise 0.

Links

Table of n, a(n) for n=-1..47.

Wolfdieter Lang, First 9 rows.

Formula

a(n, m) = 0 if n < m-1;

a(n, m) = 1 if m = 0 or n = -1;

a(n, m) = (Product_{p=0..m-1} (n-p)!)/superfac(m-1) if n >= 0, 1 <= m <= n+1, where superfac(n) := A000178(n), n >= 0, (superfactorials).

Equals ConvOffsStoT transform of the factorials, A000142: (1, 1, 2, 6, 24, ...); e.g., ConvOffs transform of (1, 1, 2, 6) = (1, 6, 12, 6, 1). - Gary W. Adamson, Apr 21 2008

Example

Rows for n = -1, 0, 1, 2, 3, ...:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  2,  1;
  1,  6, 12,  6,  1;
  ...

Crossrefs

Column sequences give: A000012 (powers of 1), A000142 (factorials), A010790, A090443-4, etc.

Cf. A090445 (row sums), A090446 (alternating row sums).

Sequence in context: A138169 A139331 A173886 * A340591 A155794 A107876

Adjacent sequences: A090438 A090439 A090440 * A090442 A090443 A090444

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Dec 23 2003

Status

approved