A364395 G.f. satisfies A(x) = 1 + x/A(x)*(1 + 1/A(x)^2).

1, 2, -8, 60, -552, 5648, -61712, 705104, -8321696, 100658368, -1241281536, 15546987648, -197234640384, 2529169695232, -32728878054144, 426864306146560, -5605439340018176, 74050470138645504, -983432207024885760, 13122261492710033408, -175836387068096147456

Offset: 0

Author

Seiichi Manyama, Jul 22 2023

Keywords

• sign

Crossrefs

Cf. A364393, A364397, A364399.

Cf. A219534.

Programs

Maple

A364395 := proc(n)
    if n = 0 then
        1;
    else
    (-1)^(n-1)*add( binomial(n,k) * binomial(2*n+2*k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364395(n),n=0..80); # R. J. Mathar, Jul 25 2023
a := n -> `if`(n=0, 1, (-1)^(n+1)*binomial(2*(n-1), n-1)*hypergeom([n-1/2, -n, n], [(n+1)/2, n/2], -1) / n):
seq(simplify(a(n)), n = 0..20);  # Peter Luschny, Mar 03 2024

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = 1 + x/A[x]*(1 + 1/A[x]^2) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)

PARI

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A219534.

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

D-finite with recurrence 9*n*(130549*n-619680) *(3*n-1)*(3*n-2)*a(n) +6*(-15361165*n^4 [phone]*n^3 -662268162*n^2 [phone]*n -435307620)*a(n-1) +4*(-908652649*n^4 [phone]*n^3 -32838390812*n^2 [phone]*n -28467674946)*a(n-2) -24*(n-3)*(50425637*n^3 -426659887*n^2 [phone]*n -890225572)*a(n-3) -16*(n-3)*(n-4) *(4607885*n -6704077)*(2*n-9)*a(n-4)=0. - R. J. Mathar, Jul 25 2023

a(n) ~ c*(-1)^(n+1)*4^n*3F2([n-1/2, -n, n], [(n+1)/2, n/2], -1)*n^(-3/2), with c = 1/(4*sqrt(Pi)). - Stefano Spezia, Oct 21 2023

a(n) = (-1)^(n+1)*binomial(2*(n-1), n-1)*hypergeom([n-1/2, -n, n], [(n+1)/2, n/2], -1) / n. - Peter Luschny, Mar 03 2024