A090441 - OEIS

%I #16 Feb 13 2022 09:25:22

%S 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,

%T 2880,120,1,1,720,86400,1036800,1036800,86400,720,1,1,5040,3628800,

%U 217728000,870912000,217728000,3628800,5040,1,1,40320,203212800,73156608000

%N Symmetric triangle of certain normalized products of decreasing factorials.

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

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

%H Wolfdieter Lang, <a href="/A090441/a090441.txt">First 9 rows</a>.

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

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

%F 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).

%F 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

%e Rows for n = -1, 0, 1, 2, 3, ...:

%e   1;

%e   1,  1;

%e   1,  1,  1;

%e   1,  2,  2,  1;

%e   1,  6, 12,  6,  1;

%e   ...

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

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

%K nonn,easy,tabl

%O -1,8

%A _Wolfdieter Lang_, Dec 23 2003