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A360271
a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * Catalan(n-3*k).
2
1, 1, 2, 5, 13, 38, 117, 373, 1222, 4085, 13877, 47766, 166229, 583893, 2067414, 7371093, 26440789, 95355990, 345538389, 1257486165, 4593933398, 16841578325, 61938532181, 228454719830, 844882459989, 3132258655573, 11638656376150, 43337083401557
OFFSET
0,3
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 = 0.2541737124933... is the smallest positive root of the equation 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
MAPLE
A360271 := proc(n)
add((-1)^k*binomial(n-3*k, k)*A000108(n-3*k), k=0..n/3) ;
end proc:
seq(A360271(n), n=0..70) ; # R. J. Mathar, Mar 12 2023
PROG
(PARI) a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
(PARI) my(N=30, x='x+O('x^N)); Vec(2/(1+(sqrt(1-4*x*(1-x^3)))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 31 2023
STATUS
approved