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A216174
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Number of Schroeder n-paths with no flat steps at ground level and equally spaced returns.
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1
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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.
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EXAMPLE
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For n=2 the 3 paths are UUDD, UFD, and UDUDUD.
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MAPLE
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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))):
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MATHEMATICA
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Table[If[n == 0, 1, Sum[(2*Hypergeometric2F1[-d + 2, d + 1, 2, -1])^(n/d), {d, Divisors[n]}]], {n, 0, 26}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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