A307045 a(n) = Sum_{k=0..floor(n/9)} (-1)^k*binomial(n,9*k).

1, 1, 1, 1, 1, 1, 1, 1, 1, 0, -9, -54, -219, -714, -2001, -5004, -11439, -24309, -48618, -92358, -167769, -292599, -490104, -783540, -1172907, -1562274, -1562274, 0, 6216183, 24581880, 72182016, 186061536, 443185425, 997483653, 2146130559, 4443424371, 8886848742

Offset: 0

Author

Seiichi Manyama, Mar 21 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 (9,-36,84,-126,126,-84,36,-9).

Crossrefs

Column 9 of A307039.

Cf. A306860.

Programs

Mathematica

a[n_] := Sum[(-1)^k * Binomial[n,9*k], {k,0,Floor[n/9]}]; Array[a, 37, 0] (* Amiram Eldar, May 25 2021 *)

PARI

{a(n) = sum(k=0, n\9, (-1)^k*binomial(n, 9*k))}

PARI

N=66; x='x+O('x^N); Vec((1-x)^8/((1-x)^9+x^9))

Formula

G.f.: (1 - x)^8/((1 - x)^9 + x^9).

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) for n > 8.