OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-3)^(n-k) * binomial(n,k) * binomial(n,k-1) for n > 0.
G.f.: 2/(1 - 4*x + sqrt(1 + 4*x + 16*x^2)).
a(n) = Sum_{k=0..n} (-3)^k * 4^(n-k) * binomial(n,k) * binomial(n+k,n)/(k+1).
(n+1) * a(n) = -2 * (2*n-1) * a(n-1) - 16 * (n-2) * a(n-2) for n>1. - Seiichi Manyama, Aug 08 2020
a(n) ~ 2^(2*n - 1/2) * ((sqrt(3) + 1)*sin(2*Pi*n/3) + (sqrt(3) - 1)*cos(2*Pi*n/3)) / (3^(3/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 04 2020
MATHEMATICA
a[0] = 1; a[n_] := Sum[(-3)^(n - k) * Binomial[n, k] * Binomial[n , k - 1], {k, 1, n}] / n; Array[a, 29, 0] (* Amiram Eldar, Aug 02 2020 *)
PROG
(PARI) {a(n) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A/(1+3*x*A)); polcoef(A, n)}
(PARI) {a(n) = if(n==0, 1, sum(k=1, n, (-3)^(n-k)*binomial(n, k)*binomial(n, k-1))/n)}
(PARI) N=40; x='x+O('x^N); Vec(2/(1-4*x+sqrt(1+4*x+16*x^2)))
(PARI) {a(n) = sum(k=0, n, (-3)^k*4^(n-k)*binomial(n, k)*binomial(n+k, n)/(k+1))}
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Aug 02 2020
STATUS
approved