A176151 Triangle T(n, k) = 2*(n+1 - mod(n, k+1) - mod(n, n-k+1) ) + (n! - k! - (n-k)!) read by rows.

1, 1, 1, 1, 6, 1, 1, 9, 9, 1, 1, 27, 26, 27, 1, 1, 105, 118, 118, 105, 1, 1, 613, 706, 714, 706, 613, 1, 1, 4333, 4930, 5016, 5016, 4930, 4333, 1, 1, 35297, 39610, 40208, 40278, 40208, 39610, 35297, 1, 1, 322577, 357856, 362168, 362742, 362742, 362168, 357856, 322577, 1

Offset

0,5

Comments

Row sums are: 1, 2, 8, 20, 82, 448, 3354, 28560, 270510, 2810688, 31841122, ...

Links

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

Formula

T(n, k) = 2*(n+1 - mod(n, k+1) - mod(n, n-k+1) ) + (n! - k! - (n-k)!).

T(n, k) = 2*A176149(n, k) - A155170(n, k). - G. C. Greubel, Feb 07 2021

Example

Triangle begins as:
  1;
  1,      1;
  1,      6,      1;
  1,      9,      9,      1;
  1,     27,     26,     27,      1;
  1,    105,    118,    118,    105,      1;
  1,    613,    706,    714,    706,    613,      1;
  1,   4333,   4930,   5016,   5016,   4930,   4333,      1;
  1,  35297,  39610,  40208,  40278,  40208,  39610,  35297,      1;
  1, 322577, 357856, 362168, 362742, 362742, 362168, 357856, 322577, 1;

Mathematica

Table[2*(n+1 -Mod[n, k+1] -Mod[n, n-k+1]) + (n! -k! -(n-k)!), {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(Sage) f=factorial; flatten([[2*(n+1 - n%(k+1) - n%(n-k+1)) + (f(n) -f(k) -f(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2021
(Magma) F:=Factorial;; [2*(n+1 - n mod (k+1) - n mod (n-k+1)) + (F(n) -F(k) -F(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2021

Crossrefs

Cf. A155170, A176149.

Sequence in context: A143087 A144470 A174377 * A204001 A363291 A144395

Adjacent sequences: A176148 A176149 A176150 * A176152 A176153 A176154

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Apr 10 2010

Extensions

Edited by G. C. Greubel, Feb 07 2021

Status

approved