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A114693 Number of returns to the x-axis in all hill-free Schroeder paths of length 2n+4. 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. 2
2, 10, 48, 232, 1138, 5666, 28592, 145984, 752978, 3918282, 20547456, 108482952, 576187554, 3076640898, 16506527392, 88938911296, 481067145826, 2611212958154, 14218923060752, 77653486423528, 425227486222482 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

FORMULA

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

a(n) = Sum_{k=0..1+floor(n/2)} k*A114692(n+2,k).

Recurrence: (n+4)*(7*n+1)*a(n) = (42*n^2 + 109*n + 49)*a(n-1) - (7*n^2 - 4*n - 5)*a(n-2) - 2*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ sqrt(15900+11243*sqrt(2))*(3+2*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

a(n) = Sum_{k=0..n+1} ( (k^3+8*k^2+15*k+8) * Sum_{j=0..n-k} ((-1)^j*2^(n+1-j)*binomial(n+1,j)*binomial(2*n-k-j,n)) )/(8*(n+1)). - Vladimir Kruchinin, Feb 27 2016.

EXAMPLE

a(0)=2 because in the three hill-free Schroeder paths of length 4, namely HH, UH(D) and UUD(D), we have altogether 2 returns to the x-axis (shown between parentheses).

MAPLE

G:=2*(1-3*z-sqrt(1-6*z+z^2))/z^2/(1+z+sqrt(1-6*z+z^2))^2: Gser:=series(G, z=0, 30): 2, seq(coeff(Gser, z^n), n=1..23);

MATHEMATICA

CoefficientList[Series[2*(1-3*x-Sqrt[1-6*x+x^2])/(x*(1+x+Sqrt[1 -6*x+x^2]))^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

PROG

(Maxima) a(n):=sum((k^3+8*k^2+15*k+8)*sum((-1)^j*2^(n+1-j)*binomial(n+1, j)*binomial(2*n-k-j, n), j, 0, n-k), k, 0, (n+1))/(8*(n+1)). /* Vladimir Kruchinin, Feb 27 2016 */

CROSSREFS

Cf. A049611, A104219, A114692.

Sequence in context: A181296 A239073 A065982 * A121950 A086853 A036918

Adjacent sequences:  A114690 A114691 A114692 * A114694 A114695 A114696

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 26 2005

STATUS

approved

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Last modified October 14 16:29 EDT 2019. Contains 328022 sequences. (Running on oeis4.)