A286033 a(n) = binomial(2*n-2, n-1) + (-1)^n.

0, 3, 5, 21, 69, 253, 923, 3433, 12869, 48621, 184755, 705433, 2704155, 10400601, 40116599, 155117521, 601080389, 2333606221, 9075135299, 35345263801, 137846528819, 538257874441, 2104098963719, 8233430727601, 32247603683099, 126410606437753, 495918532948103

Offset: 1

Author

Peter Luschny, May 13 2017

Keywords

• nonn
• easy

Comments

An odd prime p divides a((p+1)/2) which gives A163210.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000984, A033999, A163210, A178904, A219539, A000032.

Programs

Magma

[Binomial(2*n-2, n-1) + (-1)^n: n in [1..30]]; // G. C. Greubel, Jul 14 2024

Maple

a := n -> binomial(2*n-2, n-1) + (-1)^n: seq(a(n), n=1..27);

Mathematica

a[n_] := Binomial[2n-2, n-1] + (-1)^n; a[Range[1,27]]

Maxima

a(n):=-sum((-1)^k*binomial(2*n,n-k)*(fib(2*k+1)+fib(2*k-1)),k,1,n); /* Vladimir Kruchinin, Jan 18 2025 */

PARI

a(n) = binomial(2*n-2, n-1) + (-1)^n \\ David A. Corneth, May 13 2017

SageMath

def A286033(n): return binomial(2*n-2, n-1) + (-1)^n
[A286033(n) for n in range(1,31)] # G. C. Greubel, Jul 14 2024

Formula

a(n) = A000984(n-1) + A033999(n). - David A. Corneth, May 13 2017

G.f.: -1 + x/sqrt(1 - 4*x) + 1/(1 + x). - Ilya Gutkovskiy, May 13 2017

D-finite with recurrence: -(n-1)*a(n) +2*(n-1)*a(n-1) +(7*n-17)*a(n-2) +2*(2*n-7)*a(n-3)=0. - R. J. Mathar, Jan 27 2020

a(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(2*n, n-k)*A000032(2*k). - Vladimir Kruchinin, Jan 18 2025