A120582 Hankel transform of Sum_{k=0..floor(n/2)} binomial(2*k, k).

1, 2, 0, 0, -16, -32, -64, -128, 0, 0, 1024, 2048, 4096, 8192, 0, 0, -65536, -131072, -262144, -524288, 0, 0, 4194304, 8388608, 16777216, 33554432, 0, 0, -268435456, -536870912, -1073741824, -2147483648, 0, 0, 17179869184, 34359738368, 68719476736, 137438953472, 0, 0, -1099511627776

Offset: 0

Author

Paul Barry, Jun 15 2006

Keywords

• easy
• sign

Comments

Hankel transform of A100066(n+1).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,4,0,-16).

Crossrefs

Cf. A100066, A120580.

Programs

Magma

I:=[1,2,0,0]; [n le 4 select I[n] else 4*Self(n-2) -16*Self(n-4): n in [1..51]]; // G. C. Greubel, Jun 08 2022

Mathematica

LinearRecurrence[{0,4,0,-16}, {1,2,0,0}, 51] (* G. C. Greubel, Jun 08 2022 *)

SageMath

def C(n): return floor(chebyshev_U(n, 1/2))
def A120582(n): return 2^n*( ((n+1)%2)*(C(n/2) - C((n-2)/2)) + (n%2)*C((n+1)/2) )
[A120582(n) for n in (0..50)] # G. C. Greubel, Jun 08 2022

Formula

a(n) = 2^n*(-sqrt(3)*cos(5*Pi*n/6 + Pi/3)/6 + (sqrt(3)/3 - 1/2)*sin(5*Pi*n/6 + Pi/3) + (sqrt(3)/3 + 1/2)*cos(Pi*n/6 + Pi/6) + sqrt(3)*sin(Pi*n/6 + Pi/6)/6).

G.f.: (1-2*x)*(1+2*x)^2/(1 - 4*x^2 + 16*x^4). - Colin Barker, Jun 27 2013

a(n) = 2^(n-1)*( (1+(-1)^n)*(ChebyshevU(n/2, 1/2) - ChebyshevU((n-2)/2, 1/2)) + (1 -(-1)^n)*ChebyshevU((n+1)/2, 1/2)). - G. C. Greubel, Jun 08 2022