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A089164 Number of steps in all Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n,0). 1
3, 19, 107, 591, 3259, 18019, 99987, 556831, 3111347, 17436915, 97981179, 551871087, 3114878571, 17613879747, 99768824355, 565962587199, 3214923140707, 18284737574611, 104110467624075, 593397580894351 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

FORMULA

a(n) = (1/n) * Sum_{k=n..2*n} k*C(n, k-n)*C(k, n-1).

G.f.: 1/2 - 1/z + (2-7*z+z^2)/(2*z*sqrt(1-6*z+z^2)).

Recurrence: 2*(n+1)*(41*n-33)*a(n) = 3*(164*n^2-27*n+11)*a(n-1) - 2*(41*n^2+174*n-374)*a(n-2) + 69*(n-3)*a(n-3). - Vaclav Kotesovec, Oct 14 2012

a(n) ~ sqrt(48+34*sqrt(2))*(3+2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

EXAMPLE

a(2)=19 because the six Schroeder paths HH,HUD,UDH,UHD,UDUD,UUDD from (0,0) to (4,0) have 19 steps (i.e., letters) altogether.

MATHEMATICA

f[n_] := Sum[k* Binomial[n, k - n] Binomial[k, n - 1], {k, n, 2 n}] /n; Array[f, 20] (* Or *)

Rest@ CoefficientList[ Series[(x - 2 + (2 - 7 x + x^2)/(Sqrt[1 - 6 x + x^2]))/(2 x), {x, 0, 20}], x] (* Robert G. Wilson v, Sep 12 2011 *)

PROG

(PARI)  x='x+O('x^66); Vec(1/2-1/x+(2-7*x+x^2)/(2*x*sqrt(1-6*x+x^2))) \\ Joerg Arndt, May 10 2013

CROSSREFS

Cf. A006318.

Sequence in context: A047029 A095120 A151539 * A072950 A240123 A130425

Adjacent sequences:  A089161 A089162 A089163 * A089165 A089166 A089167

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 06 2003

STATUS

approved

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Last modified November 18 02:54 EST 2017. Contains 294840 sequences.