A159855 Riordan array ((1-2*x-x^2)/(1-x), x/(1-x)).

1, -1, 1, -2, 0, 1, -2, -2, 1, 1, -2, -4, -1, 2, 1, -2, -6, -5, 1, 3, 1, -2, -8, -11, -4, 4, 4, 1, -2, -10, -19, -15, 0, 8, 5, 1, -2, -12, -29, -34, -15, 8, 13, 6, 1, -2, -14, -41, -63, -49, -7, 21, 19, 7, 1, -2, -16, -55, -104, -112, -56, 14, 40, 26, 8, 1

Offset: 0

Author

Philippe Deléham, Apr 24 2009

Keywords

• easy
• sign
• tabl

Comments

The matrix inverse starts

1;

1, 1;

2, 0, 1;

2, 2, -1, 1;

4, 0, 3, -2, 1;

4, 4, -3, 5, -3, 1;

8, 0, 7, -8, 8, -4, 1;

8, 8, -7, 15, -16, 12, -5, 1;

16, 0, 15, -22, 31, -28, 17, -6, 1;

16, 16, -15, 37, -53, 59, -45, 23, -7, 1;

32, 0, 31, -52, 90, -112, 104, -68, 30, -8, 1;

- R. J. Mathar, Mar 29 2013

Examples

			Triangle begins:
   1;
  -1,  1;
  -2,  0,  1;
  -2, -2,  1,  1;
  -2, -4, -1,  2,  1;
  -2, -6, -5,  1,  3,  1;
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..5151

Crossrefs

Cf. A159854.

Programs

GAP

Flat(List([0..12],n->List([0..n],k->Binomial(n,n-k)-2*Binomial(n-1,n-k-1)-Binomial(n-2,n-k-2)))); # Muniru A Asiru, Mar 22 2018

Magma

/* As triangle */ [[Binomial(n, n-k)-2*Binomial(n-1, n-k-1)-Binomial(n-2, n-k-2): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Mar 22 2018

Maple

C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if end proc:
for n from 0 to 10 do
  seq(C(n, n-k)-2*C(n-1, n-k-1)-C(n-2, n-k-2), k = 0..n)
end do; # Peter Bala, Mar 20 2018

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - 2 # - #^2)/(1 - #)&, #/(1 - #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Sage

# uses[riordan_array from A256893]
riordan_array((1-2*x-x^2)/(1-x), x/(1-x), 8) # Peter Luschny, Mar 21 2018

Formula

T(n,k) = C(n,n-k) - 2*C(n-1,n-k-1) - C(n-2,n-k-2), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0. - Peter Bala, Mar 20 2018

Extensions

Two data values in row 10 corrected by Peter Bala, Mar 20 2018