A335997 Triangle read by rows: T(n,k) = Product_{i=n-k+1..n} i! for 0 <= k <= n.

1, 1, 1, 1, 2, 2, 1, 6, 12, 12, 1, 24, 144, 288, 288, 1, 120, 2880, 17280, 34560, 34560, 1, 720, 86400, 2073600, 12441600, 24883200, 24883200, 1, 5040, 3628800, 435456000, 10450944000, 62705664000, 125411328000, 125411328000

Offset: 0

Author

Werner Schulte, Jul 08 2020

Keywords

• nonn
• easy
• tabl

Comments

Based on some integer sequence a(n), n>0, define triangular arrays A(a;n,k) by recurrence: A(a;0,0) = 1, and A(a;i,j) = 0 if j<0 or j>i, and A(a;n,k) = n! / (n-k)! * A(a;n-1,k) + a(n) * A(a;n-1,k-1) for 0<=k<=n. Then, Product_{i=1..n} (1 + (a(i) / i!) * x) = Sum_{k=0..n} A(a;n,k) / T(n,k) * x^k for n>=0 with empty product 1 (case n=0).

For the row reversed triangle R(n,k) = Product_{i=k+1..n} i! with empty product 1 (case k=n) the terms of the matrix inverse M are given by M(n,n) = 1 for n >= 0 and M(n,n-1) = -n! for n > 0 otherwise 0. - Werner Schulte, Oct 25 2022

Examples

			The triangle starts:
n\k :  0     1      2        3         4         5         6
============================================================
  0 :  1
  1 :  1     1
  2 :  1     2      2
  3 :  1     6     12       12
  4 :  1    24    144      288       288
  5 :  1   120   2880    17280     34560     34560
  6 :  1   720  86400  2073600  12441600  24883200  24883200
  etc.
		

Crossrefs

Cf. A000012 (col_0), A000142 (col_1), A010790 (col_2), A176037 (col_3), A000178 (main diagonal and first subdiagonal).

Row sums equal A051399(n+1).

Cf. A009963, A093002, A193520.

Programs

Mathematica

T[n_, k_] := Product[i!, {i, n - k + 1, n}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 08 2020 *)

Formula

T(n,k) = T(n,1) * T(n-1,k-1) for 0 < k <= n.

T(2*n,n) = A093002(n+1) for n >= 0.

T(n,k)/T(k,k) = A009963(n,k) for 0 <= k <= n.

(Sum_{k=0..n} T(n,k) * T(n,n-k))/T(n,n) = A193520(n) for n >= 0.