A146078 Expansion of 1/(1-x*(1-9*x)).

1, 1, -8, -17, 55, 208, -287, -2159, 424, 19855, 16039, -162656, -307007, 1156897, 3919960, -6492113, -41771753, 16657264, 392603041, 242687665, -3290739704, -5474928689, 24141728647, 73416086848, -143859470975, -804604252607

Offset: 0

Author

Philippe Deléham, Oct 27 2008

Keywords

• sign
• easy

Comments

Row sums of Riordan array (1, x(1-9x)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,-9).

Crossrefs

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978.

Programs

Magma

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2018

Mathematica

LinearRecurrence[{1, -9}, {1, 1}, 100] (* G. C. Greubel, Jan 30 2016 *)

PARI

x='x+O('x^30); Vec(1/(1-x+9*x^2)) \\ G. C. Greubel, Jan 19 2018

Sage

[lucas_number1(n,1,9) for n in range(1, 27)] # Zerinvary Lajos, Apr 22 2009

Formula

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

a(n) = Sum_{k=0..n} A109466(n,k)*9^(n-k).

From G. C. Greubel, Jan 31 2016: (Start)

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

E.g.f.: exp(x/2)*(cos(sqrt(35)*x/2) + (1/sqrt(35))*sin(sqrt(35)*x/2)). (End)

a(n) = Product_{k=1..n} (1 + 6*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019

a(n) = 3^n * U(n, 1/6), where U(n, x) is the Chebyshev polynomial of the second kind. - Federico Provvedi, Mar 28 2022