OFFSET
0,2
COMMENTS
The offset is chosen following the Deleham offset in A124576, not according to the less systematic offset in the definition.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
FORMULA
Conjecture: 3*n*a(n) +2*(-13*n+9)*a(n-1) +4*(13*n-21)*a(n-2) +24*(-n+2)*a(n-3)=0.
a(n) ~ 2^(n-3/2)*3^(n+1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Jul 06 2013
G.f.: 1/(6*x -1 +2*sqrt((2*x-1)*(6*x-1))). - Vaclav Kotesovec, Jul 06 2013
MAPLE
AA := proc(n, k, x, y)
option remember;
if k <0 or k > n then
0 ;
elif n = 0 then
1;
elif k = 0 then
x*procname(n-1, k, x, y)+procname(n-1, 1, x, y) ;
else
procname(n-1, k-1, x, y)+y*procname(n-1, k, x, y)+procname(n-1, k+1, x, y) ;
end if;
end proc:
seq(add( AA(n, k, 1, 4), k=0..n), n=0..30) ;
MATHEMATICA
CoefficientList[Series[1/(6*x-1+2*Sqrt[(2*x-1)*(6*x-1)]), {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 06 2013 *)
PROG
(PARI) x='x+O('x^30); Vec(1/(6*x -1 +2*sqrt((2*x-1)*(6*x-1)))) \\ G. C. Greubel, Nov 19 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/(6*x -1 +2*Sqrt((2*x-1)*(6*x-1))) )); // G. C. Greubel, Nov 19 2018
(Sage) s= (1/(6*x -1 +2*sqrt((2*x-1)*(6*x-1)))).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 19 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Jun 30 2013
STATUS
approved