A089383 Number of peaks at even level in all Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the axis) from (0,0) to (2n+4,0).
1, 8, 49, 280, 1569, 8752, 48833, 272976, 1529441, 8589176, 48342449, 272640680, 1540495553, 8718956768, 49423735553, 280551815456, 1594568513857, 9073566717800, 51686272315569, 294711466792120, 1681938025818081, 9606920311565328, 54915241962566849, 314131983462253680
Offset: 0
Keywords
Examples
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).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Crossrefs
Cf. A006318.
Programs
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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 *) -
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
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
Comments