A117413 Expansion of (1-x^2)/(1-2*x^2+4*x^3+x^4).

1, 0, 1, -4, 1, -12, 17, -24, 81, -104, 241, -508, 817, -1876, 3425, -6512, 13537, -24848, 49697, -97332, 185249, -368604, 710129, -1380872, 2709425, -5233656, 10232209, -19924140, 38689617, -75543460, 146843585, -285921248, 557171393, -1083673376, 2111184193, -4110111076

Offset: 0

Author

Paul Barry, Mar 14 2006

Keywords

• easy
• sign

Comments

Diagonal sums of number triangle A117411.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,2,-4,-1).

Crossrefs

Cf. A117411.

Programs

Magma

I:=[1,0,1,-4]; [n le 4 select I[n] else 2*Self(n-1) -4*Self(n-2) -Self(n-3): n in [1..41]]; // G. C. Greubel, Sep 07 2022

Mathematica

CoefficientList[Series[(1-x^2)/(1-2x^2+4x^3+x^4),{x,0,40}],x] (* or *) LinearRecurrence[{0,2,-4,-1},{1,0,1,-4},40] (* Harvey P. Dale, Jul 12 2017 *)

SageMath

@CachedFunction
def a(n): # a = A117413
    if(n<4): return (1, 0, 1, -4)[n]
    else: return 2*a(n-2) - 4*a(n-3) - a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Sep 07 2022

Formula

a(n) = 2*a(n-2) - 4*a(n-3) - a(n-4).

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2*k} C(n-k, k-j)*C(j, n-2*k)*(-4)^(n-2*k).

a(n) = (-)^n*A052992(2*n). - R. J. Mathar, Nov 22 2024