A387649 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(2*k+1,2*n-4*k).
1, 0, 1, 6, 1, 20, 21, 42, 141, 128, 505, 782, 1465, 3852, 5501, 14290, 26053, 49464, 113313, 195446, 433585, 841732, 1629285, 3477754, 6520445, 13583472, 26955209, 52921182, 109117737, 212250492, 431217101, 861537698, 1706182325, 3460479528, 6839829361
Offset: 0
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).
Programs
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Magma
[&+[2^(n-2*k)* Binomial(2*k+1, 2*n-4*k): k in [0..Floor (n/2)]]: n in [0..40]]; // Vincenzo Librandi, Sep 06 2025
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Mathematica
Table[Sum[2^(n-2*k)*Binomial[2*k+1,2*n-4*k],{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 06 2025 *) -
PARI
a(n) = sum(k=0, n\2, 2^(n-2*k)*binomial(2*k+1, 2*n-4*k));
Formula
G.f.: (1-x^2+2*x^3)/((1-x^2+2*x^3)^2 - 8*x^3).
a(n) = 2*a(n-2) + 4*a(n-3) - a(n-4) + 4*a(n-5) - 4*a(n-6).