A216174 Number of Schroeder n-paths with no flat steps at ground level and equally spaced returns.
1, 1, 3, 7, 27, 91, 439, 1807, 9059, 41803, 214231, 1037719, 5460691, 27297739, 145340511, 746123815, 4011076915, 20927156707, 113608631567, 600318853927, 3279271467435, 17524510115443, 96226513851535, 518431875418927, 2861594917241083, 15521473553775091
Offset: 0
Keywords
Examples
For n=2 the 3 paths are UUDD, UFD, and UDUDUD.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..400
Programs
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Maple
b:= n-> coeff(series((1-x-(1-6*x+x^2)^(1/2))/(2*x), x, n+3), x, n): a:= n-> `if`(n=0, 1, add(b(d-1)^(n/d), d=numtheory[divisors](n))): seq(a(n), n=0..30); # Alois P. Heinz, Sep 13 2012
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Mathematica
Table[If[n == 0, 1, Sum[(2*Hypergeometric2F1[-d + 2, d + 1, 2, -1])^(n/d), {d, Divisors[n]}]], {n, 0, 26}]
Formula
a(0)=1, a(n) = Sum_{d|n} (2*hypergeom([-d+2, d+1], [2], -1))^(n/d) = Sum_{d|n} A006318(d-1)^(n/d) for n >=1.