OFFSET
0,2
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 +161422948*n^3 -662268162*n^2 +955427047*n -435307620)*a(n-1) +4*(-908652649*n^4 +9061174176*n^3 -32838390812*n^2 +51018866685*n -28467674946)*a(n-2) -24*(n-3)*(50425637*n^3 -426659887*n^2 +1128823867*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
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 *)
PROG
(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);
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jul 22 2023
STATUS
approved