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A107415
Triangle, read by rows: T(0,0) = 1; T(n,k) = n!*T(n-1,k) - T(n-1,k-1).
2
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
0,4
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
LINKS
Seiichi Manyama, Rows n = 0..43, flattened
EXAMPLE
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
PROG
(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
CROSSREFS
KEYWORD
easy,sign,tabl
AUTHOR
Gerald McGarvey, May 26 2005
STATUS
approved