A165620 Riordan array ((1-x)/(1-x^4),x/(1+x^2)).

1, -1, 1, 0, -1, 1, 0, -1, -1, 1, 1, 1, -2, -1, 1, -1, 2, 2, -3, -1, 1, 0, -2, 4, 3, -4, -1, 1, 0, -2, -4, 7, 4, -5, -1, 1, 1, 2, -6, -7, 11, 5, -6, -1, 1, -1, 3, 6, -13, -11, 16, 6, -7, -1, 1, 0, -3, 9, 13, -24, -16, 22, 7, -8, -1, 1

Offset

0,13

Comments

Diagonal sums are (-1)^n. Row sums have g.f. 1/(1+x^3).

The transform of the aerated Catalan numbers by this matrix is (-1)^n.

The transform of the shifted central binomial coefficient C(n+1,floor((n+1)/2)) is 1^n.

Factorizes as (1/(1+x),x)*(1/(1+x^2),x/(1+x^2)). Inverse is A165621.

Links

Table of n, a(n) for n=0..65.

P. Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.

Formula

Number triangle T(n,k)=sum{j=0..n, (-1)^(n-j)(-1)^((j-k)/2)(1+(-1)^(j-k))C((j+k)/2,k)/2}.

Example

Triangle begins
1,
-1, 1,
0, -1, 1,
0, -1, -1, 1,
1, 1, -2, -1, 1,
-1, 2, 2, -3, -1, 1,
0, -2, 4, 3, -4, -1, 1,
0, -2, -4, 7, 4, -5, -1, 1,
1, 2, -6, -7, 11, 5, -6, -1, 1,
-1, 3, 6, -13, -11, 16, 6, -7, -1, 1

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - #^4)&, #/(1 + #^2)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Crossrefs

Sequence in context: A271325 A228431 A328702 * A128494 A257696 A110730

Adjacent sequences: A165617 A165618 A165619 * A165621 A165622 A165623

Keyword

easy,sign,tabl

Author

Paul Barry, Sep 22 2009

Status

approved