A375372 Expansion of 1/( (1 + x) * (1 - x^2*(1 + x)^2) ).

1, -1, 2, 0, 2, 2, 5, 5, 12, 16, 28, 44, 73, 115, 190, 304, 494, 798, 1293, 2089, 3384, 5472, 8856, 14328, 23185, 37511, 60698, 98208, 158906, 257114, 416021, 673133, 1089156, 1762288, 2851444, 4613732, 7465177, 12078907, 19544086, 31622992, 51167078, 82790070

Offset: 0

Author

Seiichi Manyama, Aug 13 2024

Keywords

• sign
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (-1,1,3,3,1).

Crossrefs

Cf. A093040, A375373.

Programs

PARI

my(N=50, x='x+O('x^N)); Vec(1/((1+x)*(1-x^2*(1+x)^2)))

PARI

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

Formula

a(n) = -a(n-1) + a(n-2) + 3*a(n-3) + 3*a(n-4) + a(n-5).

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

a(n) = A375373(n) + A375373(n-1).

2*a(n) = 2*(-1)^n + A000045(n) + A057078(n+1). - R. J. Mathar, Aug 14 2024