OFFSET
0,2
COMMENTS
Apparently, number of Dyck (n+3)-paths with no descent having the same length as the preceding ascent. - David Scambler, Apr 28 2012 (Proved by S. Elizalde, Disc. Math., 2021)
REFERENCES
S. Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, Discrete Math., 344 (2021), no. 6, 112364.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020, see Theorem 2.1 for t=0 and r=1.
FORMULA
G.f.: (1-3*x+x^2-x^3-(1-x)*sqrt(1-4*x+2*x^2+x^4))/(2*x^4).
Conjecture: (n+4)*a(n)-(4*n+9)*a(n-1) +(2*n-1)*a(n-2) -a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Nov 17 2011
a(n) = Sum_{m=1..(n+2)/2} C(2*m,m)/(m+1)*C(n+m+1,3*m-1)). - Vladimir Kruchinin, Jan 24 2022
EXAMPLE
For n=1, Dyck 4-paths are (2,-1,2,-3), (3,-1,1,-3) and (3,-2,1,-2), a(1) = 3.
MATHEMATICA
CoefficientList[Series[(1-3*x+x^2-x^3-(1-x)*Sqrt[1-4*x+2*x^2+x^4])/( 2*x^4), {x, 0, 30}], x] (* G. C. Greubel, Nov 16 2018 *)
PROG
(PARI) x='x+O('x^30); Vec((1-3*x+x^2-x^3-(1-x)*sqrt(1-4*x+2*x^2 +x^4))/( 2*x^4)) \\ G. C. Greubel, Nov 16 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-3*x+x^2-x^3-(1-x)*Sqrt(1-4*x+2*x^2 +x^4))/( 2*x^4) ));
(Sage) s=((1-3*x+x^2-x^3-(1-x)*sqrt(1-4*x+2*x^2+x^4))/( 2*x^4)).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 16 2018
(Maxima)
a(n):=sum((binomial(2*m, m)*binomial(n+m+1, 3*m-1))/(m+1), m, 1, (n+2)/2); /* Vladimir Kruchinin, Jan 24 2022 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Apr 01 2011
STATUS
approved