A387603 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^(n-k) * binomial(2*n-4*k+2,2*k+1).

1, 4, 12, 40, 160, 640, 2416, 8960, 33664, 127744, 484096, 1827840, 6896896, 26049536, 98440192, 371939328, 1404997632, 5307301888, 20049424384, 75742707712, 286136467456, 1080936235008, 4083451559936, 15426119532544, 58275554459648, 220148448624640, 831657574400000

Offset: 0

Author

Seiichi Manyama, Sep 02 2025

Keywords

• nonn
• easy
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-4,8,16,0,-16).

Crossrefs

Cf. A387509.

Programs

Magma

[&+[2^(n-k)* Binomial(2*n-4*k+2, 2*k+1)/2: k in [0..Floor (n/3)]]: n in [0..35]]; // Vincenzo Librandi, Sep 03 2025

Mathematica

Table[Sum[2^(n-k)*Binomial[2*n-4*k+2, 2*k+1]/2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 03 2025 *)

PARI

a(n) = sum(k=0, n\3, 2^(n-k)*binomial(2*n-4*k+2, 2*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387509.

G.f.: 1/((1-2*x-4*x^3)^2 - 32*x^4).

a(n) = 4*a(n-1) - 4*a(n-2) + 8*a(n-3) + 16*a(n-4) - 16*a(n-6).