OFFSET
1,4
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
FORMULA
G.f. (for offset 0): (1+4x-sqrt(1-8x+12x^2))/(2x^2+8x).
G.f. as continued fraction is 1/(1-0*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(.....
a(s) = Sum_{n=1..s}( Sum_{k=0..floor((s-2*n)/2)} 4^(s-2*n-2*k)*(n/(n+2*k))*binomial(n+2*k, k)*binomial(s-n-1, s-2*n-2*k) ) with s>=2.
D-finite with recurrence: 4*n*a(n) +(48-31n)*a(n-1) +4*(10n-33)*a(n-2) +12*(n-3)*a(n-3)=0. - R. J. Mathar, Jan 27 2012
a(n) ~ 3 * 6^(n-1/2) / (25*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jan 31 2014
a(n) = 1/(n+1)*Sum_{j=0..floor(n/2)} 4^(n-2*j)*C(n+1,j)*C(n-j-1,n-2*j). - Vladimir Kruchinin, Apr 04 2019
MAPLE
with(LREtools): with(FormalPowerSeries): # requires Maple 2022
ogf:= (1+4*x-sqrt(1-8*x+12*x^2))/(2*x^2+8*x):
init:= [1, 0, 1, 4, 18, 84, 405, 2004];
iseq:= seq(u(i-1)=init[i], i=1..nops(init)): req:= FindRE(ogf, x, u(n));
rmin:= subs(n=n-4, MinimalRecurrence(req, u(n), {iseq})[1]); # Mathar's recurrence
a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
seq(a(n), n=0..24); # Georg Fischer, Nov 03 2022
MATHEMATICA
CoefficientList[Series[(1+4*x-Sqrt[1-8*x+12*x^2])/(2*x^2+8*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 31 2014 *)
PROG
(PARI) x='x+O('x^50); Vec((1+4*x-sqrt(1-8*x+12*x^2))/(2*x^2+8*x)) \\ G. C. Greubel, Jun 23 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
José Luis Ramírez Ramírez, Jan 26 2012
STATUS
approved