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A216174 Number of Schroeder n-paths with no flat steps at ground level and equally spaced returns.

%I A216174 #28 May 27 2015 22:01:01
%S A216174 1,1,3,7,27,91,439,1807,9059,41803,214231,1037719,5460691,27297739,
%T A216174 145340511,746123815,4011076915,20927156707,113608631567,600318853927,
%U A216174 3279271467435,17524510115443,96226513851535,518431875418927,2861594917241083,15521473553775091
%N A216174 Number of Schroeder n-paths with no flat steps at ground level and equally spaced returns.
%H A216174 Alois P. Heinz, <a href="/A216174/b216174.txt">Table of n, a(n) for n = 0..400</a>
%F A216174 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.
%e A216174 For n=2 the 3 paths are UUDD, UFD, and UDUDUD.
%p A216174 b:= n-> coeff(series((1-x-(1-6*x+x^2)^(1/2))/(2*x), x, n+3), x, n):
%p A216174 a:= n-> `if`(n=0, 1, add(b(d-1)^(n/d), d=numtheory[divisors](n))):
%p A216174 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 13 2012
%t A216174 Table[If[n == 0, 1, Sum[(2*Hypergeometric2F1[-d + 2, d + 1, 2, -1])^(n/d), {d, Divisors[n]}]], {n, 0, 26}]
%Y A216174 Cf. A006318, A216435.
%K A216174 nonn
%O A216174 0,3
%A A216174 _David Scambler_, Sep 13 2012