A364371 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^2).

1, 0, -1, 2, -2, -1, 9, -20, 20, 24, -150, 327, -293, -599, 3097, -6452, 4854, 15878, -71252, 140112, -81328, -437346, 1746254, -3214989, 1223971, 12345295, -44552833, 76242173, -11292089, -354175849, 1167638037, -1842585992, -233903034, 10273377388, -31169512310

Offset: 0

Author

Seiichi Manyama, Jul 20 2023

Keywords

• sign
• easy

Crossrefs

Cf. A364372, A364373, A364375.

Cf. A000108, A073157.

Programs

Maple

A364371 := proc(n)
    add((-1)^k* binomial(2*k+1,k) * binomial(2*k+1,n-k)/(2*k+1),k=0..n) ;
end proc:
seq(A364371(n),n=0..70); # R. J. Mathar, Jul 25 2023

PARI

a(n) = sum(k=0, n, (-1)^k*binomial(2*k+1, k)*binomial(2*k+1, n-k)/(2*k+1));

Formula

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

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

D-finite with recurrence (n+1)*a(n) +(5*n-1)*a(n-1) +6*(2*n-3)*a(n-2) +6*(2*n-5)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 25 2023

From Peter Bala, Aug 24 2024: (Start)

A(x) = (1 + x)*c(-x*(1+x)^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.

(1/x) * series_reversion(x/A(x)) = 1 - x^2 + 2*x^3 - 11*x^5 + 28*x^6 + ..., the g.f. of A364375. (End)