A360153 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-6*k,n-3*k).
1, 2, 6, 21, 72, 258, 945, 3504, 13128, 49565, 188260, 718560, 2753721, 10588860, 40835160, 157871241, 611669250, 2374441380, 9233006541, 35956933050, 140220970200, 547490880981, 2140055896770, 8373651697800, 32795094564081, 128550662334522
Offset: 0
Keywords
Programs
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Maple
A360153 := proc(n) add(binomial(2*n-6*k,n-3*k),k=0..n/3) ; end proc: seq(A360153(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
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Mathematica
a[n_] := Sum[Binomial[2*n - 6*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-6*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)))
Formula
G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3) ).
a(n) ~ 2^(2*n + 6) / (63 * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
a(n)-a(n-3) = A000984(n). - R. J. Mathar, Mar 12 2023
D-finite with recurrence n*a(n) +2*(-2*n+1)*a(n-1) -n*a(n-3) +2*(2*n-1)*a(n-4)=0. - R. J. Mathar, Mar 12 2023