A104538 Expansion of g.f. (1 + 2*x) / (1 + 2*x + 4*x^2).

1, 0, -4, 8, 0, -32, 64, 0, -256, 512, 0, -2048, 4096, 0, -16384, 32768, 0, -131072, 262144, 0, -1048576, 2097152, 0, -8388608, 16777216, 0, -67108864, 134217728, 0, -536870912, 1073741824, 0, -4294967296, 8589934592, 0, -34359738368, 68719476736, 0, -274877906944, 549755813888, 0

Offset: 0

Author

Paul Barry, Mar 13 2005

Keywords

• easy
• sign

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,-4).

Crossrefs

Cf. A057078, A287479.

Programs

Mathematica

CoefficientList[Series[(1+2x)/(1+2x+4x^2),{x,0,50}],x] (* or *) LinearRecurrence[ {-2,-4},{1,0},50] (* Harvey P. Dale, Sep 18 2022 *)

PARI

a(n) = 2^n*polchebyshev(2*n, 2, 1/2); \\ Michel Marcus, Aug 05 2018

PARI

Vec((1 + 2*x) / (1 + 2*x + 4*x^2) + O(x^40)) \\ Colin Barker, Aug 08 2018

Formula

a(n) = 2^n*A057078(n), where A057078(n) = U(2n, 1/2), U(n, x) Chebyshev polynomial of second kind.

a(n) = b(n) + b(n+1) where b(n) = 0, 1, -1, -3, 11, -11, -21, 85, -85, ... is the inverse binomial transform of A287479(n). - Paul Curtz, Aug 05 2018

a(n) = (((-1+i*sqrt(3))^n*(-i+sqrt(3)) + (-1-i*sqrt(3))^n*(i+sqrt(3)))) / (2*sqrt(3)) where i=sqrt(-1). - Colin Barker, Aug 08 2018

E.g.f.: exp(-x)*(sqrt(3)*cos(sqrt(3)*x) + sin(sqrt(3)*x))/sqrt(3). - Stefano Spezia, Jul 15 2024