OFFSET
0,2
COMMENTS
Partial sums of A026002.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
G.f.: (1-z-q)^2/(4*z^2*(1-z)*q), where q = sqrt(1-6*z+z^2).
Recurrence: (n+2)*n^2*a(n) = (n+1)*(7*n^2+4*n+1)*a(n-1) - (7*n^2+10*n+4)*n * a(n-2) + (n-1)*(n+1)^2*a(n-3). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ sqrt(1632+1154*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 24 2012
EXAMPLE
a(0) = 1 because the paths HH, HUD, UDH, UHD, UDUD and U(UD)D from (0,0) to (4,0) have only one peak at an even level (shown between parentheses).
MATHEMATICA
CoefficientList[Series[(1-x-Sqrt[1-6*x+x^2])^2/(4*x^2*(1-x)* Sqrt[1-6*x+x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)
PROG
(PARI) x='x+O('x^66); q = sqrt(1-6*x+x^2); Vec((1-x-q)^2/(4*x^2*(1-x)*q)) \\ Joerg Arndt, May 10 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 28 2003
STATUS
approved