A216174 - OEIS

%I #28 May 27 2015 22:01:01

%S 1,1,3,7,27,91,439,1807,9059,41803,214231,1037719,5460691,27297739,

%T 145340511,746123815,4011076915,20927156707,113608631567,600318853927,

%U 3279271467435,17524510115443,96226513851535,518431875418927,2861594917241083,15521473553775091

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

%H Alois P. Heinz, <a href="/A216174/b216174.txt">Table of n, a(n) for n = 0..400</a>

%F 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 For n=2 the 3 paths are UUDD, UFD, and UDUDUD.

%p b:= n-> coeff(series((1-x-(1-6*x+x^2)^(1/2))/(2*x), x, n+3), x, n):

%p a:= n-> `if`(n=0, 1, add(b(d-1)^(n/d), d=numtheory[divisors](n))):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 13 2012

%t Table[If[n == 0, 1, Sum[(2*Hypergeometric2F1[-d + 2, d + 1, 2, -1])^(n/d), {d, Divisors[n]}]], {n, 0, 26}]

%Y Cf. A006318, A216435.

%K nonn

%O 0,3

%A _David Scambler_, Sep 13 2012