A360313 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-1-k,n-2*k) * binomial(2*k,k).

1, 0, -2, -2, 4, 10, -4, -38, -22, 114, 188, -234, -914, -18, 3376, 3338, -9416, -21718, 14416, 96338, 39274, -328558, -471344, 795398, 2586064, -517690, -10453424, -8272658, 32186818, 63596494, -61876584, -307070174, -62655330, 1129250706, 1356328788

Offset: 0

Author

Seiichi Manyama, Feb 03 2023

Keywords

• sign

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A360314, A360315.

Cf. A026585, A360293.

Programs

Mathematica

RecurrenceTable[{a[0]==1,a[1]==0,a[2]==-2,a[n]==1/n (2(n-1)a[n-1]-(5n-6)a[n-2]+2(2n-5)a[n-3])},a,{n,40}] (* Harvey P. Dale, Sep 20 2024 *)

PARI

a(n) = sum(k=0, n\2, (-1)^k*binomial(n-1-k, n-2*k)*binomial(2*k, k));

PARI

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1+4*x^2/(1-x)))

Formula

G.f.: 1 / sqrt(1+4*x^2/(1-x)).

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