A106509 Riordan array ((1+x)/(1+x+x^2), x/(1+x)), read by rows.

1, 0, 1, -1, -1, 1, 1, 0, -2, 1, 0, 1, 2, -3, 1, -1, -1, -1, 5, -4, 1, 1, 0, 0, -6, 9, -5, 1, 0, 1, 0, 6, -15, 14, -6, 1, -1, -1, 1, -6, 21, -29, 20, -7, 1, 1, 0, -2, 7, -27, 50, -49, 27, -8, 1, 0, 1, 2, -9, 34, -77, 99, -76, 35, -9, 1, -1, -1, -1, 11, -43, 111, -176, 175, -111, 44, -10, 1

Offset: 0

Author

Paul Barry, May 04 2005

Keywords

• easy
• sign
• tabl

Comments

Row sums are A106510.

Diagonal sums are A106511.

Inverse of A072405 (when this starts 1, 0, 1, ...).

Examples

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

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Crossrefs

Cf. A072405, A106510, A106511.

Programs

Magma

T:= func< n,k | (&+[ (-1)^j*Binomial(2*n-k-j, j): j in [0..n-k]]) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 28 2021

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 + #)/(1 + # + #^2)&, #/(1 + #)&, 12] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Sage

def T(n,k): return sum( (-1)^j*binomial(2*n-k-j, j) for j in (0..n-k))
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2021

Formula

T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(2n-k-j, j).

T(n,k) = T(n-1,k-1) - 2*T(n-1,k) + T(n-2,k-1) - 2*T(n-2,k) + T(n-3,k-1) - T(n-3,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = 0, T(2,1) = T(2,0) = -1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 12 2014

Sum_{k=0..n} T(n,k) = A106510(n). - G. C. Greubel, Apr 28 2021