A383582 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * binomial(2*(n-4*k),n-4*k).
1, 2, 6, 20, 71, 256, 942, 3512, 13221, 50138, 191260, 733088, 2821037, 10892100, 42174848, 163706656, 636816019, 2481902842, 9689155902, 37882580356, 148313102097, 581365577564, 2281393560802, 8961689897248, 35235582858441, 138657185501870, 546064549476476
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
Programs
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Magma
[&+[Binomial(n-3*k,k) * Binomial(2*(n-4*k),n-4*k): k in [0..n div 4]]: n in [0..45]]; // Vincenzo Librandi, May 02 2025
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Mathematica
Table[Sum[Binomial[n-3*k,k]* Binomial[2*(n-4*k),n-4*k],{k,0,Floor[n/4]}],{n,0,30}] (* Vincenzo Librandi, May 02 2025 *) -
PARI
a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*(n-4*k), n-4*k));
Formula
G.f.: 1/sqrt((1 - x^4) * (1 - x^4 - 4*x)).
a(n) ~ (2 + sqrt(2) + sqrt(10 + 8*sqrt(2)))^n / (sqrt((sqrt(5 + 32*sqrt(2)) - 7)*Pi*n) * 2^(n + 7/4)). - Vaclav Kotesovec, May 01 2025