A360271 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-3*k,k) * Catalan(n-3*k).

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

Links

Table of n, a(n) for n=0..27.

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\3, (-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

Cf. A157003.

Cf. A000108, A360219, A360272.

Sequence in context: A148302 A149857 A001475 * A343937 A369729 A149858

Adjacent sequences: A360268 A360269 A360270 * A360272 A360273 A360274

Keyword

nonn

Author

Seiichi Manyama, Jan 31 2023

Status

approved