A110524 Expansion of (1 + x)/(1 + 2*x + 2*x^3).

1, -1, 2, -6, 14, -32, 76, -180, 424, -1000, 2360, -5568, 13136, -30992, 73120, -172512, 407008, -960256, 2265536, -5345088, 12610688, -29752448, 70195072, -165611520, 390727936, -921846016, 2174915072, -5131286016, 12106264064, -28562358272, 67387288576, -158987105280

Offset: 0

Author

Paul Barry, Jul 24 2005

Keywords

• easy
• sign

Comments

Diagonal sums of number triangle A110522.

Links

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

Crossrefs

Cf. A077999.

Programs

GAP

a:=[1,-1,2];; for n in [4..40] do a[n]:=-2*(a[n-1]+a[n-3]); od; a; # G. C. Greubel, Jun 27 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)/( 1+2*x+2*x^3) )); // G. C. Greubel, Jun 27 2019

Mathematica

CoefficientList[Series[(1+x)/(1+2*x+2*x^3), {x,0,40}], x] (* G. C. Greubel, Aug 30 2017 *)
LinearRecurrence[{-2,0,-2}, {1,-1,2}, 40] (* G. C. Greubel, Jun 27 2019 *)

PARI

my(x='x+O('x^40)); Vec((1+x)/(1+2*x+2*x^3)) \\ G. C. Greubel, Aug 30 2017

Sage

((1+x)/(1+2*x+2*x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..(n-k)} (-1)^(n-k-j)*C(n-k, j) *(-3)^(j-k)*C(k, j-k).

a(n) = (-1)^n * A077999(n). - G. C. Greubel, Jun 27 2019