A155170 Triangle T(n, k) = k! - n! + (n-k)! read by rows.

1, 1, 1, 1, 0, 1, 1, -3, -3, 1, 1, -17, -20, -17, 1, 1, -95, -112, -112, -95, 1, 1, -599, -694, -708, -694, -599, 1, 1, -4319, -4918, -5010, -5010, -4918, -4319, 1, 1, -35279, -39598, -40194, -40272, -40194, -39598, -35279, 1, 1, -322559, -357838, -362154, -362736, -362736, -362154, -357838, -322559, 1

Offset

0,8

Comments

Row sums are: 1, 2, 2, -4, -52, -412, -3292, -28492, -270412, -2810572, -31840972, ...

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Formula

Sum_{k=0..n} T(n, m) = 2*A003422(n+1) - A000142(n+1). - R. J. Mathar, Jun 24 2011

Example

Triangle begins as:
  1;
  1,       1;
  1,       0,       1;
  1,      -3,      -3,       1;
  1,     -17,     -20,     -17,       1;
  1,     -95,    -112,    -112,     -95,       1;
  1,    -599,    -694,    -708,    -694,    -599,       1;
  1,   -4319,   -4918,   -5010,   -5010,   -4918,   -4319,       1;
  1,  -35279,  -39598,  -40194,  -40272,  -40194,  -39598,  -35279,       1;
  1, -322559, -357838, -362154, -362736, -362736, -362154, -357838, -322559, 1;

Mathematica

Table[k! -n! +(n-k)!, {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Sage) f=factorial; flatten([[f(n-k) -f(n) +f(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2021
(Magma) F:=Factorial;; [F(n-k) -F(n) +F(k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2021

Crossrefs

Cf. A000142, A003422, A176151, A176152.

Sequence in context: A075837 A178885 A087107 * A126460 A173503 A338114

Adjacent sequences: A155167 A155168 A155169 * A155171 A155172 A155173

Keyword

sign,tabl,easy

Author

Roger L. Bagula, Jan 21 2009

Status

approved