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A114710 Number of hill-free Schroeder paths of length 2n that have no horizontal steps on the x-axis. 3
1, 0, 2, 6, 26, 114, 526, 2502, 12194, 60570, 305526, 1560798, 8058714, 41987106, 220470942, 1165553718, 6198683090, 33140219946, 178012804678, 960232902606, 5199384505226, 28250295397170, 153977094874862, 841656387060006 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

LINKS

Table of n, a(n) for n=0..23.

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.

FORMULA

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

Apparently 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

EXAMPLE

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

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);

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 *)

CROSSREFS

Column 0 of A114709.

Cf. A104219.

Sequence in context: A192435 A296217 A050890 * A230245 A288606 A208034

Adjacent sequences:  A114707 A114708 A114709 * A114711 A114712 A114713

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 26 2005

STATUS

approved

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Last modified February 19 05:12 EST 2018. Contains 299330 sequences. (Running on oeis4.)