A386301 a(n) = 1 + Sum_{k=0..n-1} (-1)^k * a(k) * a(n-1-k).

1, 2, 1, -1, 1, 8, 1, -28, 1, 134, 1, -649, 1, 3320, 1, -17497, 1, 94526, 1, -520507, 1, 2910896, 1, -16487794, 1, 94393106, 1, -545337199, 1, 3175320608, 1, -18615098836, 1, 109783526822, 1, -650884962907, 1, 3877184797784, 1, -23193307022860, 1, 139271612505362

Offset: 0

Author

Seiichi Manyama, Jul 17 2025

Keywords

• sign

Crossrefs

Cf. A007317, A348857, A348858, A348859.

Cf. A064641, A386300.

Programs

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=1+sum(j=0, i-1, (-1)^j*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies:

(1) A(x) = 1/( (1-x) * (1-x*A(-x)) ).

(2) A(x)*A(-x) = B(-x^2), where B(x) is the g.f. of A064641.

(3) A(x) = 1/(1-x) + 2*x/(1+x^2 + sqrt(1+6*x^2-3*x^4)).

a(2*n) = 1 and a(2*n+1) = 1 + (-1)^n * A064641(n) for n >= 0.