OFFSET
0,3
REFERENCES
JiSun Huh and SeungKyung Park, The Chung-Feller Theorem on Generalized Lattice Paths, submitted.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = (1/(n-1))*Sum_{k=1..n-1} binomial(n-1, k)*(binomial(n, k-1)+binomial(n-1, k-2))*4^(n-k) for n > 1.
a(n) = Sum_{k=1..n-1} C_{k+1}*C(n-2,k-1)*2^k where C_n is n-th Catalan number.
G.f.: (1-x)*(1+3*x-sqrt(9*x^2-10*x+1))/(8*x).
a(n) ~ 3^(2*n-1)/(sqrt(Pi/2)*n^(3/2)). - Vaclav Kotesovec, Jan 28 2013
Conjecture: (n+1)*a(n) +2*(-5*n+3)*a(n-1) +9*(n-3)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
MAPLE
gf := (1-x)*(1+3*x-sqrt(9*x^2-10*x+1))/(8*x): s := series(gf, x, 100): for i from 0
to 50 do printf(‘%d, ’, coeff(s, x, i)) od:
MATHEMATICA
CoefficientList[Series[(1-x)*(1+3*x-Sqrt[9*x^2-10*x+1])/(8*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 28 2013 *)
PROG
(PARI) x = 'x + O('x^66); Vec((1-x)*(1+3*x-sqrt(9*x^2-10*x+1))/(8*x)) /* Joerg Arndt, Jan 23 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Jisun Huh, Jan 23 2013
STATUS
approved