A360290 a(n) = Sum_{k=0..floor(n/2)} binomial(n-1-k,k) * binomial(2*n-4*k,n-2*k).
1, 2, 6, 22, 82, 314, 1222, 4814, 19138, 76626, 308550, 1248230, 5069266, 20654602, 84392838, 345659166, 1418769154, 5834283298, 24031706246, 99134911542, 409495076050, 1693539077210, 7011618614342, 29058701620974, 120540377731266, 500443750830962
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
Programs
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Magma
[&+[Binomial(n-1-k, k) * Binomial(2*n-4*k, n-2*k): k in [0..Floor(n div 2)]]: n in [0..30]]; // Vincenzo Librandi, May 04 2025
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Mathematica
Table[Sum[Binomial[n-1-k,k]* Binomial[2*n-4*k, n-2*k],{k,0,Floor[n/2]}],{n,0,35}] (* Vincenzo Librandi, May 04 2025 *) -
PARI
a(n) = sum(k=0, n\2, binomial(n-1-k, k)*binomial(2*n-4*k, n-2*k));
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PARI
my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^2)))
Formula
G.f.: 1 / sqrt(1-4*x/(1-x^2)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(n-2)*a(n-2) - 2*(2*n-7)*a(n-3) - (n-4)*a(n-4).
a(n) ~ phi^(3*n) / (5^(1/4) * sqrt(Pi*n/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Feb 02 2023