A360151 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k,n-3*k).
1, 2, 6, 21, 74, 267, 981, 3648, 13690, 51744, 196699, 751237, 2880345, 11080081, 42743148, 165291569, 640563158, 2487083484, 9672626600, 37674470433, 146937686295, 573781535775, 2243050091905, 8777451670102, 34379401083017, 134770951530840
Offset: 0
Keywords
Programs
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Maple
A360151 := proc(n) add(binomial(2*n-4*k,n-3*k),k=0..n/3) ; end proc: seq(A360151(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
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Mathematica
a[n_] := Sum[Binomial[2*n - 4*k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *) -
PARI
a(n) = sum(k=0, n\3, binomial(2*n-4*k, n-3*k));
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PARI
my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^2)))
Formula
G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^2) ), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+4) / (15*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
D-finite with recurrence +2*n*a(n) +(-11*n+6)*a(n-1) +(19*n-24)*a(n-2) +2*(-16*n+33)*a(n-3) +2*(11*n-36)*a(n-4) +(-25*n+78)*a(n-5) +6*(n-3)*a(n-6) +4*(-2*n+9)*a(n-7)=0. - R. J. Mathar, Mar 12 2023