OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = C(2*n,n) - (-1)^(n-1)*Sum_{i=0..[(n-1)/2]} C(3*i,i)*C(i-n-1,n-1-2*i)/(2*i+1).
From Vaclav Kotesovec, May 01 2018: (Start)
Recurrence: 2*(n-1)*n*(2*n + 1)*(5*n - 6)*a(n) = (n-1)^2*(115*n^2 - 138*n + 56)*a(n-1) + 4*(n-2)*(n+1)*(2*n - 3)*(5*n - 11)*a(n-2) - 36*(n-2)*(2*n - 5)*(2*n - 3)*(5*n - 1)*a(n-3).
a(n) ~ 4^n / (phi^2 * sqrt(Pi*n)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. (End)
MATHEMATICA
a[n_] := Binomial[2*n, n] - (-1)^(n-1)*Sum[ Binomial[3*k, k]*Binomial[k - n-1, n-1-2*k]/(2*k+1), {k, 0, Floor[(n-1)/2]}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 30 2018 *)
PROG
(PARI) {a(n)=binomial(2*n, n)+(-1)^n*sum(i=0, (n-1)\2, binomial(3*i, i) *binomial(i-n-1, n-1-2*i)/(2*i+1))}
(Magma) [1] cat [Binomial(2*n, n) - (-1)^(n-1)*(&+[Binomial(3*k, k)*Binomial(k-n - 1, n-2*k-1)/(2*k+1): k in [0..Floor((n-1)/2)]]): n in [1..50]]; // G. C. Greubel, Apr 30 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 06 2007
STATUS
approved