OFFSET
0,2
COMMENTS
Sum_{n>=0} (-a(n)/(-4)^n) is the Cauchy product of Sum_{n>=0} (-A349844(n)/(-8)^n) with itself.
LINKS
Wikipedia, Cauchy product.
FORMULA
For n > 0, a(n) = 8*binomial(2*(n-1),n-1) - binomial(2*n,n) = binomial(2*(n-1),n-1) * (4 + 2/n).
a(n) ~ 4^n * (1/sqrt(Pi*n)).
From Amiram Eldar, Mar 13 2026: (Start)
Sum_{n>=0} 1/a(n) = 4/3 - 10*Pi/(9*sqrt(3)).
Sum_{n>=0} (-1)^(n+1)/a(n) = (4/5)*(11*log(phi)/sqrt(5) - 1), where phi is the golden ratio (A001622). (End)
EXAMPLE
a(1) = binomial(0,0) * (4 + 2/1) = 6;
a(2) = binomial(2,1) * (4 + 2/2) = 10;
a(3) = binomial(4,2) * (4 + 2/3) = 28;
a(4) = binomial(6,3) * (4 + 2/4) = 90.
MATHEMATICA
a[n_] := Binomial[2*(n-1), n-1]*(4 + 2/n); a[0] = -1; Array[a, 30, 0] (* Amiram Eldar, Mar 13 2026 *)
PROG
(PARI) a(n) = if(n, binomial(2*(n-1), n-1) * (4 + 2/n), -1)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Jianing Song, Dec 01 2021
STATUS
approved