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A216174
Number of Schroeder n-paths with no flat steps at ground level and equally spaced returns.
1
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
0,3
LINKS
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.
EXAMPLE
For n=2 the 3 paths are UUDD, UFD, and UDUDUD.
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
MATHEMATICA
Table[If[n == 0, 1, Sum[(2*Hypergeometric2F1[-d + 2, d + 1, 2, -1])^(n/d), {d, Divisors[n]}]], {n, 0, 26}]
CROSSREFS
Sequence in context: A098465 A148750 A081562 * A260464 A346658 A246313
KEYWORD
nonn
AUTHOR
David Scambler, Sep 13 2012
STATUS
approved