A364393 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^2).

1, 2, -4, 20, -120, 800, -5696, 42416, -326304, 2572992, -20685696, 168920704, -1397257472, 11682707712, -98578346496, 838369268480, -7178912946688, 61842549386240, -535575159363584, 4660216874719232, -40722264390799360, 357204260381327360

Offset: 0

Author

Seiichi Manyama, Jul 22 2023

Keywords

• sign

Crossrefs

Cf. A364395, A364397, A364399.

Cf. A346626.

Programs

Maple

A364393 := proc(n)
    if n = 0 then
        1;
    else
        (-1)^(n-1)*add( binomial(n,k) * binomial(n+2*k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364393(n),n=0..70); # R. J. Mathar, Jul 25 2023

Mathematica

m = 22;
A[_] = 1;
Do[A[x_] = 1 + x*(1 + 1/A[x]^2) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Sep 05 2023 *)

PARI

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A346626.

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

D-finite with recurrence 2*n*(2*n-1)*a(n) +(35*n^2-64*n+24) *a(n-1) +(-35*n^2+205*n-288) *a(n-2) +2*(-43*n^2+341*n-660) *a(n-3) -4*(7*n-30)*(n-5) *a(n-4) -8*(n-5)*(n-6)*a(n-5)=0. - R. J. Mathar, Jul 25 2023

a(n) = (-1)^(n-1)*n*3F2([1-n, (n+1)/2, n/2+1], [3/2, 2], -1) for n > 1. - Stefano Spezia, Oct 21 2023