A307089 Expansion of (1 - x)^4/((1 - x)^6 + x^6).

1, 2, 3, 4, 5, 6, 6, 0, -27, -110, -319, -780, -1702, -3404, -6315, -10864, -17051, -23238, -23238, 0, 87021, 325358, 890077, 2107560, 4542526, 9085052, 16950573, 29354524, 46296905, 63239286, 63239286, 0, -236031147, -880918070, -2406788599, -5694626340

Offset: 0

Author

Seiichi Manyama, Mar 24 2019

Keywords

• sign
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-2).

Crossrefs

Column 6 of A307079.

Row sums of triangle A307156.

Cf. A119336.

Programs

Mathematica

a[n_] := Sum[(-1)^k * Binomial[n+1, 6*k+1], {k, 0, Floor[n/6]}]; Array[a, 36, 0] (* Amiram Eldar, May 14 2021 *)

PARI

{a(n) = sum(k=0, n\6, (-1)^k*binomial(n+1,6*k+1))}

PARI

N=66; x='x+O('x^N); Vec((1-x)^4/((1-x)^6+x^6))

Formula

a(n) = Sum_{k=0..floor(n/6)} (-1)^k*binomial(n+1,6*k+1).

a(n) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i,3*j) * binomial(n-i,3*j).

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - 2*a(n-6) for n > 5.