A384494 Triangle read by rows: T(n, k) = (-1)^k*(k+1)*(n+1-k)!, n >= 0, k = 0..n.

1, 2, -2, 6, -4, 3, 24, -12, 6, -4, 120, -48, 18, -8, 5, 720, -240, 72, -24, 10, -6, 5040, -1440, 360, -96, 30, -12, 7, 40320, -10080, 2160, -480, 120, -36, 14, -8, 362880, -80640, 15120, -2880, 600, -144, 42, -16, 9, 3628800, -725760, 120960, -20160, 3600, -720, 168, -48, 18, -10

Offset: 0

Author

Wolfdieter Lang, May 31 2025

Keywords

• sign
• tabl
• easy

Comments

This triangle, written as (infinite) square matrix MT with vanishing upper diagonals 0, together with the Riordan triangle A104698, written also as such a square matrix MR, appears in the double sum formula for the number of certain restricted permutations given in A086852(n), as diagonal sequence A086852(n+2) = (2*MR*MT^t)_{n,n}, for n >=0, where t indicates matrix transpositon.

Examples

			The triangle T begins:
  n\k        0        1       2       3     4     5   6   7  8    9 ...
  ---------------------------------------------------------------------
  0:         1
  1:         2       -2
  2:         6       -4       3
  3:        24      -12       6      -4
  4:       120      -48      18      -8     5
  5:       720     -240      72     -24    10    -6
  6:      5040    -1440     360     -96    30   -12   7
  7:     40320   -10080    2160    -480    12   -36  14  -8
  8:    362880   -80640   15120   -2880   600  -144  42 -16  9
  9:   3628800  -725760  120960  -20160  3600  -720 168 -48 18 -10
  ...
		

Crossrefs

Cf. A104698, A086852, A086852.

Column sequences: A000142(n+1), -A052849, A052560(n-1), -A052578(n-2), A052648(n-3), -A298881(n-4), A062098(n-5), -A159038(n-6), ...

Programs

Mathematica

Table[(-1)^k * (k+1) * (n+1-k)!, {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 31 2025 *)

Formula

T(n, k) = (-1)^k*(k+1)*(n+1-k)!, for n >= 0 and k = 0, 1, ..., n.

O.g.f. of row polynomials P(n, y) := Sum_{k=0..n} T(n, k) y^k: G(x, y) = ((N(x) - 1)/x) * (1/(1 + y*x)^2), with N(x) = hypergeometric([1,1], [], x), the o.g.f. of {n!}_{n>=0} (see A000142).