A307395 Expansion of 1/((1 - x) * ((1 - x)^3 + x^3)).

1, 4, 10, 19, 28, 28, 1, -80, -242, -485, -728, -728, 1, 2188, 6562, 13123, 19684, 19684, 1, -59048, -177146, -354293, -531440, -531440, 1, 1594324, 4782970, 9565939, 14348908, 14348908, 1, -43046720, -129140162, -258280325, -387420488, -387420488, 1, 1162261468

Offset: 0

Author

Seiichi Manyama, Apr 07 2019

Keywords

• sign
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..4000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,3).

Crossrefs

Column 5 of A307394.

Partial sums of A057083.

Programs

Mathematica

LinearRecurrence[{4, -6, 3}, {1, 4, 10}, 38] (* Amiram Eldar, May 13 2021 *)

PARI

{a(n) = sum(k=0, n\3, (-1)^k*binomial(n+3, 3*k+3))}

PARI

N=66; x='x+O('x^N); Vec(1/((1-x)*((1-x)^3+x^3)))

Formula

a(n) = Sum_{k=0..floor(n/3)} (-1)^k*binomial(n+3,3*k+3).

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

a(6*n) = 1.

a(n) = 1 - A057681(n+3). - Yomna Bakr and Greg Dresden, Apr 22 2024