A306940 Expansion of 1/((1 - x)^6 + x^6).

1, 6, 21, 56, 126, 252, 461, 780, 1209, 1638, 1638, 0, -6187, -23238, -63783, -151316, -326382, -652764, -1217483, -2107560, -3322995, -4538430, -4538430, 0, 16942381, 63239286, 172791861, 408855776, 880983606, 1761967212, 3287837741, 5694626340

Offset: 0

Author

Seiichi Manyama, Mar 17 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 A306914.

Programs

Mathematica

CoefficientList[Series[1/((1 - x)^6 + x^6), {x, 0, 31}], x] (* Amiram Eldar, May 25 2021 *)
LinearRecurrence[{6,-15,20,-15,6,-2},{1,6,21,56,126,252},40] (* Harvey P. Dale, May 31 2021 *)

PARI

{a(n) = sum(k=0, n\6, (-1)^k*binomial(n+5, 6*k+5))}

PARI

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

Formula

a(n) = Sum_{k=0..floor(n/6)} (-1)^k*binomial(n+5,6*k+5).

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.