A087169 Expansion of (1 + 3*x)/(1 + 5*x + 9*x^2).

1, -2, 1, 13, -74, 253, -599, 718, 1801, -15467, 61126, -166427, 282001, 87838, -2977199, 14095453, -43682474, 91553293, -64624199, -500858642, 3085911001, -10921827227, 26835937126, -35883240587, -62107231199, 633485321278, -2608461525599, 7340939736493, -13228544952074

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 22 2003

Keywords

• easy
• sign

Comments

For positive n, a(n) equals 3^n times the permanent of the (2n) X (2n) tridiagonal matrix with 1/sqrt(3)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (where i is the imaginary unit). - John M. Campbell, Jul 08 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-5,-9).

Crossrefs

Cf. A190970.

Programs

Magma

[n le 2 select (-1)^(n-1)*n else -5*Self(n-1) - 9*Self(n-2): n in [1..51]]; // G. C. Greubel, Jun 09 2022

Mathematica

CoefficientList[Series[(1+3x)/(1+5x+9x^2), {x, 0, 30}], x]
LinearRecurrence[{-5,-9},{1,-2},50] (* Harvey P. Dale, Mar 24 2021 *)

SageMath

def A087169(n): return (-3)^n*(chebyshev_U(n, 5/6) - chebyshev_U(n-1, 5/6))
[A087169(n) for n in (0..50)] # G. C. Greubel, Jun 09 2022

Formula

G.f.: (1 + 3*x)/(1 + 5*x + 9*x^2).

a(n) = -5*a(n-1) - 9*a(n-2), a(0)=1, a(1)=-2.

a(n) = Sum_{k=0..n} binomial(n+k,2*k)*(-3)^(n-k).

a(n) = (-3)^n*( ChebyshevU(n, 5/6) - ChebyshevU(n-1, 5/6) ). - G. C. Greubel, Jun 09 2022