A114710 Number of hill-free Schroeder paths of length 2n that have no horizontal steps on the x-axis.

1, 0, 2, 6, 26, 114, 526, 2502, 12194, 60570, 305526, 1560798, 8058714, 41987106, 220470942, 1165553718, 6198683090, 33140219946, 178012804678, 960232902606, 5199384505226, 28250295397170, 153977094874862, 841656387060006

Offset: 0

Author

Emeric Deutsch, Dec 26 2005

Keywords

• nonn
• easy

Comments

A Schroeder path of length 2n is a lattice path from (0, 0) to (2n, 0) consisting of U = (1,1), D = (1,-1) and H = (2,0) steps and never going below the x-axis. A hill is a peak at height 1.

Hankel transform is 2^C(n+1,2) (A006125(n+1)). Hankel transform of a(n+1) is (2-2^(n+1))*2^C(n+1,2). - Paul Barry, Oct 31 2008

Examples

			a(3) = 6 because we have UHHD, UHUDD, UUDHD, UUDUDD, UUHDD and UUUDDD.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1000
• Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
• Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.

Crossrefs

Column 0 of A114709.

Cf. A001003, A104219, A297705.

Programs

Maple

G:=2/(1+3*z+sqrt(1-6*z+z^2)): Gser:=series(G,z=0,32):
1,seq(coeff(Gser,z^n),n=1..27);
# Alternative:
a := proc(n) option remember; if n < 3 then return [1, 0, 2, 6][n+1] fi;
((4 - 2*n)*a(n-3) + (16*n - 11)*a(n-1) + 9*n*a(n-2))/(3*n + 3) end:
seq(a(n), n = 0..23); # Peter Luschny, Nov 10 2022

Mathematica

A114710[n_] := (-1)^n Sum[Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, 2], {k, 0, n}]; Table[A114710[n], {n, 0, 23}] (* Peter Luschny, Jan 08 2018 *)
InverseInvertTransform[ser_, n_] := CoefficientList[Series[ser/(1 + x ser), {x, 0, n}], x]; LittleSchroeder := (1 + x - Sqrt[1 - 6 x + x^2])/(4 x); (* A001003 *)
InverseInvertTransform[LittleSchroeder, 23] (* Peter Luschny, Jan 10 2019 *)

Formula

G.f.: A(x) = 2/(1+3*x+sqrt(1-6*x+x^2)).

D-finite with recurrence 3*(n+1)*a(n) +(11-16*n)*a(n-1) -9*n*a(n-2) +2*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 07 2012

G.f.: 1/(Q(0) + 2*x) where Q(k) = 1 + k*(1-x) - x - x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013

a(n) = (-1)^n*Sum_{k=0..n} binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], 2). - Peter Luschny, Jan 08 2018

O.g.f. A(x) = 1/x * series reversion of x*(1 - 3*x)/((1 - x)*(1 - 2*x)). Cf. A297705. - Peter Bala, Nov 08 2022

a(n) ~ (9 + 4*sqrt(2)) * (1 + sqrt(2))^(2*n + 1) / (49 * sqrt(Pi) * 2^(3/4) * n^(3/2)). - Vaclav Kotesovec, Nov 10 2022