A187307 Hankel transform of an alternating sum of Motzkin numbers.

1, 2, 2, -1, -5, -5, 1, 8, 8, -1, -11, -11, 1, 14, 14, -1, -17, -17, 1, 20, 20, -1, -23, -23, 1, 26, 26, -1, -29, -29, 1, 32, 32, -1, -35, -35, 1, 38, 38, -1, -41, -41, 1, 44, 44, -1, -47, -47, 1, 50, 50, -1, -53, -53, 1, 56, 56, -1, -59, -59, 1

Offset: 0

Author

Paul Barry, Mar 08 2011

Keywords

• sign
• easy

Comments

Hankel transform of A187306.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..9999
• Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
• Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 16.
• Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1)

Crossrefs

Cf. A187306.

Programs

Mathematica

LinearRecurrence[{2,-3,2,-1},{1,2,2,-1},120] (* Harvey P. Dale, Jan 21 2019 *)

PARI

Vec((1+x^2-x^3)/(1-x+x^2)^2 + O(x^80)) \\ Michel Marcus, Sep 26 2017

Formula

G.f.: (1+x^2-x^3)/(1-x+x^2)^2.

a(n) = (1/3)*(2*n+2-(n-1)*cos(2*n*Pi/3)-(n-1)*cos(4*n*Pi/3)-cos((2*n+1)*Pi/3)-sin((8*n+1)*Pi/6))*(-1)^floor(n/3). - Wesley Ivan Hurt, Sep 25 2017