A387651 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(2*k+1,2*n-4*k+1).

1, 0, 3, 2, 5, 20, 11, 70, 93, 176, 515, 618, 1877, 3212, 6043, 14670, 23469, 55336, 104915, 201202, 442213, 801924, 1711787, 3369814, 6574205, 13748896, 26354723, 54064250, 107762933, 212616060, 434126907, 852642910, 1721793229, 3443972120, 6839443699, 13808517634

Offset: 0

Author

Seiichi Manyama, Sep 05 2025

Keywords

• nonn
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1500
• Index entries for linear recurrences with constant coefficients, signature (0,2,4,-1,4,-4).

Crossrefs

Cf. A387627, A387652.

Cf. A387647, A387649.

Programs

Magma

[&+[2^(n-2*k)* Binomial(2*k+1, 2*n-4*k+1): k in [0..Floor (n/2)]]: n in [0..40]]; // Vincenzo Librandi, Sep 05 2025

Mathematica

Table[Sum[2^(n-2*k)*Binomial[2*k+1,2*n-4*k+1],{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 05 2025 *)

PARI

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

Formula

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

a(n) = 2*a(n-2) + 4*a(n-3) - a(n-4) + 4*a(n-5) - 4*a(n-6).