A376787 Expansion of (1 - x^2 + x^3)/((1 - x^2 + x^3)^2 - 4*x^3).

1, 0, 1, 3, 1, 10, 6, 21, 36, 43, 127, 139, 340, 540, 881, 1832, 2653, 5427, 8829, 15550, 28642, 46805, 87756, 147575, 262751, 465591, 797864, 1437816, 2471553, 4383696, 7689305, 13402819, 23752217, 41305842, 72916606, 127708213, 223809012, 394045411

Offset: 0

Author

Seiichi Manyama, Oct 04 2024

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,2,-1).

Crossrefs

Cf. A376716, A376788.

Cf. A376723, A376726, A376729.

Programs

Mathematica

CoefficientList[Series[(1-x^2+x^3)/((1-x^2+x^3)^2-4x^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,2,2,-1,2,-1},{1,0,1,3,1,10},40] (* Harvey P. Dale, Aug 11 2025 *)

PARI

my(N=40, x='x+O('x^N)); Vec((1-x^2+x^3)/((1-x^2+x^3)^2-4*x^3))

PARI

a(n) = sum(k=0, n\2, binomial(2*k+1, 2*n-4*k));

Formula

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).

a(n) = Sum_{k=0..floor(n/2)} binomial(2*k+1,2*n-4*k).