A360049 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+2,3*k+2) * Catalan(k).

1, 3, 6, 9, 9, 0, -26, -72, -117, -82, 204, 975, 2289, 3357, 1332, -9834, -37935, -82593, -108282, 2583, 487521, 1621071, 3261546, 3685230, -2318615, -24607854, -72887472, -134909701, -123941901, 200330184, 1258932996, 3377359872, 5706502677, 3797618237

Offset

0,2

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Formula

a(n) = binomial(n+2,2) - Sum_{k=0..n-3} a(k) * a(n-k-3).

G.f. A(x) satisfies: A(x) = 1/(1-x)^3 - x^3 * A(x)^2.

G.f.: 2 / ( (1-x) * ((1-x)^2 + sqrt((1-x)^4 + 4*x^3*(1-x))) ).

D-finite with recurrence (n+3)*a(n) +4*(-n-2)*a(n-1) +6*(n+1)*a(n-2) -6*a(n-3) +3*(-n+1)*a(n-4)=0. - R. J. Mathar, Jan 25 2023

Mathematica

Table[Sum[(-1)^k Binomial[n+2, 3k+2]CatalanNumber[k], {k, 0, Floor[n/3]}], {n, 0, 40}] (* Harvey P. Dale, Nov 21 2023 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(n+2, 3*k+2)*binomial(2*k, k)/(k+1));
(PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)*((1-x)^2+sqrt((1-x)^4+4*x^3*(1-x)))))

Crossrefs

Cf. A360048, A360050, A360051.

Cf. A000108.

Sequence in context: A021077 A114041 A212712 * A325750 A325748 A057083

Adjacent sequences: A360046 A360047 A360048 * A360050 A360051 A360052

Keyword

sign

Author

Seiichi Manyama, Jan 23 2023

Status

approved