A169660 Triangle, read by rows, T(n, k) = binomial(n-1,k-1)*n!/k! + binomial(n-1, n-k)* n!/(n-k+1)! - n!.

1, 1, 1, 1, 6, 1, 1, 24, 24, 1, 1, 140, 120, 140, 1, 1, 1110, 780, 780, 1110, 1, 1, 10122, 8190, 3360, 8190, 10122, 1, 1, 100856, 101976, 30240, 30240, 101976, 100856, 1, 1, 1088712, 1332576, 512064, 60480, 512064, 1332576, 1088712, 1, 1, 12700890

Offset

1,5

Comments

Row sums are: {1, 2, 8, 50, 402, 3782, 39986, 466146, 5927186, 81594182, ...}.

The first negative terms are T(11,6) = -11975040, T(12,6) = T(12,7) = -127733760, T(13,7) = -3943779840, T(14,7) = T(14,8) = -53785892160. - Hugo Pfoertner, Jul 16 2020

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Formula

T(n, k) = binomial(n-1,k-1)*n!/k! + binomial(n-1, n-k)*n!/(n-k+1)! - n!.

Example

Triangle begins as:
  1;
  1,       1;
  1,       6,       1;
  1,      24,      24,      1;
  1,     140,     120,    140,     1;
  1,    1110,     780,    780,  1110,      1;
  1,   10122,    8190,   3360,  8190,  10122,       1;
  1,  100856,  101976,  30240, 30240, 101976,  100856,       1;
  1, 1088712, 1332576, 512064, 60480, 512064, 1332576, 1088712, 1;

Maple

b:=binomial; seq(seq( b(n-1, k-1)*n!/k! + b(n-1, n-k)*n!/(n-k+1)! -n!, k=1..n), n=1..10); # G. C. Greubel, Nov 28 2019

Mathematica

T[n_, k_]:= Binomial[n-1, k-1]*n!/k! +Binomial[n-1, n-k]*n!/(n-k+1)! -n!; Table[T[n, m], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Nov 28 2019 *)

Prog

(PARI) T(n, k) = binomial(n-1, k-1)*n!/k! + binomial(n-1, n-k)*n!/(n-k+1)! - n!; \\ G. C. Greubel, Nov 28 2019
(Magma) F:=Factorial; B:=Binomial; [B(n-1, k-1)*F(n)/F(k) + B(n-1, n-k)*F(n)/F(n - k+1) - F(n): k in [1..n], n in [1..10]]; // G. C. Greubel, Nov 28 2019
(Sage) f=factorial; b=binomial; [[b(n-1, k-1)*f(n)/f(k) + b(n-1, n-k)*f(n)/f(n - k+1) - f(n) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Nov 28 2019
(GAP) F:=Factorial;; B:=Binomial;; Flat(List([1..10], n-> List([1..n], k->
B(n-1, k-1)*F(n)/F(k) + B(n-1, n-k)*F(n)/F(n - k+1) - F(n) ))); # G. C. Greubel, Nov 28 2019

Crossrefs

Sequence in context: A155863 A173882 A174045 * A035348 A140945 A141688

Adjacent sequences: A169657 A169658 A169659 * A169661 A169662 A169663

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 05 2010

Status

approved