OFFSET
1,2
COMMENTS
A Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
FORMULA
G.f.: (1-x)*(1-x-sqrt(1-6*x+x^2))/(2*sqrt(1-6*x+x^2)).
a(n) = Jacobi_P(n+1,-1,-2,3). [Paul Barry, Sep 27 2009]
Recurrence: n*a(n) = (7*n-6)*a(n-1) - (7*n-22)*a(n-2) + (n-4)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(6*sqrt(2)-8)*(3+2*sqrt(2))^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012
a(n) = Hyper2F1([-n, n-1], [1], -1). - Peter Luschny, Aug 02 2014
a(n) = Sum_{k=0..n+1} C(n+1,k)*C(n+k-1,k). - Vladimir Kruchinin, Jun 15 2020
EXAMPLE
a(2)=4 because we have (HH),(H)UD,UD(H),U(H)D,UDUD and UUDD; the 4 horizontal segments are shown between parentheses.
MAPLE
G:=(1-z)*(1-z-sqrt(1-6*z+z^2))/2/sqrt(1-6*z+z^2): Gser:=series(G, z=0, 28): seq(coeff(Gser, z^n), n=1..24);
a := n -> hypergeom([-n, n-1], [1], -1);
seq(round(evalf(a(n), 36)), n=1..23); # Peter Luschny, Aug 02 2014
MATHEMATICA
Rest[CoefficientList[Series[(1-x)*(1-x-Sqrt[1-6*x+x^2])/ (2*Sqrt[1 -6*x+x^2]), {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 17 2012 *)
PROG
(PARI) x='x+O('x^66); Vec((1-x)*(1-x-sqrt(1-6*x+x^2))/(2*sqrt(1-6*x+x^2))) \\ Joerg Arndt, May 13 2013
(Maxima) a(n):=sum(binomial(n+1, k)*binomial(n+k-1, k), k, 0, n+1) /* Vladimir Kruchinin, Jun 15 2020 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 14 2005
STATUS
approved