A360048 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+1,2*k+1) * Catalan(k).

1, 2, 2, 0, -3, -2, 9, 24, 11, -66, -152, -8, 587, 1082, -438, -5248, -7733, 7942, 47502, 53792, -105313, -430118, -343043, 1249800, 3866557, 1730018, -13996096, -34243896, -1947203, 150962374, 296101865, -121857184, -1582561869, -2468098042, 2529520766

Offset

0,2

Links

Table of n, a(n) for n=0..34.

Formula

a(n) = n+1 - Sum_{k=0..n-2} a(k) * a(n-k-2).

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

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

D-finite with recurrence (n+2)*a(n) +3*(-n-1)*a(n-1) +(7*n-4)*a(n-2) +5*(-n+1)*a(n-3)=0. - R. J. Mathar, Jan 25 2023

Prog

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

Crossrefs

Cf. A360049, A360050, A360051.

Cf. A000108.

Sequence in context: A327028 A271707 A341445 * A127899 A128615 A087508

Adjacent sequences: A360045 A360046 A360047 * A360049 A360050 A360051

Keyword

sign

Author

Seiichi Manyama, Jan 23 2023

Status

approved