A156045 Triangle T(n, k) = 2 + n! - k! - (n-k)!, read by rows.

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 19, 22, 19, 1, 1, 97, 114, 114, 97, 1, 1, 601, 696, 710, 696, 601, 1, 1, 4321, 4920, 5012, 5012, 4920, 4321, 1, 1, 35281, 39600, 40196, 40274, 40196, 39600, 35281, 1, 1, 322561, 357840, 362156, 362738, 362738, 362156, 357840, 322561, 1

Offset

0,5

Comments

Row sums are: {1, 2, 4, 12, 62, 424, 3306, 28508, 270430, 2810592, 31840994, ...}.

Links

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

Formula

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

Sum_{k=0..n} T(n,k) = 2*(n+1) + (n+1)! - 2*!(n+1), where !n = A003422(n). - G. C. Greubel, Dec 02 2019

Example

Triangle begins as:
  1;
  1,     1;
  1,     2,     1;
  1,     5,     5,     1;
  1,    19,    22,    19,     1;
  1,    97,   114,   114,    97,     1;
  1,   601,   696,   710,   696,   601,     1;
  1,  4321,  4920,  5012,  5012,  4920,  4321,     1;
  1, 35281, 39600, 40196, 40274, 40196, 39600, 35281, 1;

Maple

seq(seq( n! -k! -(n-k)! +2, k=0..n), n=0..10); # G. C. Greubel, Dec 02 2019

Mathematica

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

Prog

(PARI) T(n, k) = n! -k! -(n-k)! +2; \\ G. C. Greubel, Dec 02 2019
(Magma) F:=Factorial; [F(n) -F(k) -F(n-k) +2: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 02 2019
(Sage) f=factorial; [[f(n) -f(k) -f(n-k) +2 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 02 2019
(GAP) F:=Factorial;; Flat(List([0..10], n-> List([0..n], k-> F(n) -F(k) -F(n-k) +2 ))); # G. C. Greubel, Dec 02 2019

Crossrefs

Sequence in context: A378173 A060854 A091378 * A119687 A086856 A052916

Adjacent sequences: A156042 A156043 A156044 * A156046 A156047 A156048

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 02 2009

Extensions

Edited by G. C. Greubel, Dec 02 2019

Status

approved