A360272 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(n-3*k).
1, 1, 2, 5, 15, 46, 147, 485, 1642, 5669, 19883, 70646, 253755, 919925, 3361546, 12368661, 45786219, 170400470, 637200555, 2392962645, 9021255722, 34128098389, 129519490219, 492966689110, 1881289209003, 7197100511317, 27595769836714, 106032318322517
Offset: 0
Keywords
Programs
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Maple
A360272 := proc(n) add(binomial(n-3*k,k)*A000108(n-3*k),k=0..n/3) ; end proc: seq(A360272(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
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PARI
a(n) = sum(k=0, n\4, binomial(n-3*k, k)*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+(sqrt(1-4*x*(1+x^3)))))
Formula
G.f.: c(x * (1+x^3)), where c(x) is the g.f. of A000108.
a(n) ~ 2 * sqrt(1-3*r) / (sqrt(Pi) * n^(3/2) * r^n), where r = 0.2463187933841... is the smallest positive root of the equation1 1 - 4*r - 4*r^4 = 0. - 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*(-4*n+11)*a(n-4) +4*(-n+5)*a(n-7)=0. - R. J. Mathar, Mar 12 2023