A107415 Triangle, read by rows: T(0,0) = 1; T(n,k) = n!*T(n-1,k) - T(n-1,k-1).

1, 1, -1, 2, -3, 1, 12, -20, 9, -1, 288, -492, 236, -33, 1, 34560, -59328, 28812, -4196, 153, -1, 24883200, -42750720, 20803968, -3049932, 114356, -873, 1, 125411328000, -215488512000, 104894749440, -15392461248, 579404172, -4514276, 5913, -1

Offset: 0

Author

Gerald McGarvey, May 26 2005

Keywords

• easy
• sign
• tabl

Comments

For n>0, the row sums are 0. For n>1, sum(k=0..n) 2^k*T(n,k) = 0. The first subdiagonal (1,-3,9,-33,...) is an alternating signed version of A007489 (sum of k!, k=1..n). The first column is A000178 (superfactorials).

Also triangle of coefficients in expansion of Product_{k=0..n} (k! - x) in ascending powers of x. - Seiichi Manyama, Sep 24 2021

Examples

			Triangle begins
         1;
         1,        -1;
         2,        -3,        1;
        12,       -20,        9,       -1;
       288,      -492,      236,      -33,      1;
     34560,    -59328,    28812,    -4196,    153,   -1;
  24883200, -42750720, 20803968, -3049932, 114356, -873, 1;
(1 - x) * (2 - x) = 2 - 3*x + x^2, (1 - x) * (2 - x) * (6 - x) = 12 - 20*x + 9*x^2 - x^3, etc. - _Seiichi Manyama_, Sep 24 2021
		

Links

• Seiichi Manyama, Rows n = 0..43, flattened

Crossrefs

Cf. A000178, A008955, A249677.

Programs

PARI

t(n, k) = {if (k < 0, return (0)); if (n < k, return (0)); if (n == 0, return (1)); return (n!*t(n-1, k) - t(n-1, k-1));} \\ Michel Marcus, Apr 11 2013

PARI

row(n) = Vecrev(prod(k=1, n, k!-x)); \\ Seiichi Manyama, Sep 24 2021