A360274 a(n) = Sum_{k=0..floor(n/3)} Catalan(n-3*k).
1, 1, 2, 6, 15, 44, 138, 444, 1474, 5000, 17240, 60260, 213012, 760140, 2734700, 9907857, 36117810, 132379490, 487546557, 1803381000, 6696499910, 24953813577, 93285944640, 349756113560, 1314857960901, 4955232346092, 18717109185712, 70848408876905
Offset: 0
Programs
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Maple
A360274 := proc(n) add(A000108(n-3*k),k=0..n/3) ; end proc: seq(A360274(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
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PARI
a(n) = sum(k=0, n\3, binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
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PARI
my(N=30, x='x+O('x^N)); Vec(2/((1-x^3)*(1+sqrt(1-4*x))))
Formula
G.f.: c(x)/(1-x^3), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+6) / (63*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 01 2023
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-n-1)*a(n-3) +2*(2*n-1)*a(n-4)=0. - R. J. Mathar, Mar 12 2023