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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 06 2003
STATUS
approved