A375373 Expansion of 1/( (1 + x)^2 * (1 - x^2*(1 + x)^2) ).

1, -2, 4, -4, 6, -4, 9, -4, 16, 0, 28, 16, 57, 58, 132, 172, 322, 476, 817, 1272, 2112, 3360, 5496, 8832, 14353, 23158, 37540, 60668, 98238, 158876, 257145, 415988, 673168, 1089120, 1762324, 2851408, 4613769, 7465138, 12078948, 19544044, 31623034, 51167036

Offset: 0

Author

Seiichi Manyama, Aug 13 2024

Keywords

• sign

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,0,4,6,4,1).

Crossrefs

Cf. A093040, A375372.

Programs

Mathematica

CoefficientList[Series[1/((1+x)^2(1-x^2(1+x)^2)),{x,0,50}],x] (* or *) LinearRecurrence[{-2,0,4,6,4,1},{1,-2,4,-4,6,-4},50] (* Harvey P. Dale, Dec 11 2024 *)

PARI

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

PARI

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

Formula

a(n) = -2*a(n-1) + 4*a(n-3) + 6*a(n-4) + 4*a(n-5) + a(n-6).

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

2*a(n) = 2*(-1)^n*(n+1) +A212804(n)-A057078(n). - R. J. Mathar, Aug 14 2024