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A212388 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 8).

%I A212388 #13 May 11 2015 19:48:17
%S A212388 1,1,1,1,1,1,1,1,1,2,11,56,221,716,2003,5006,11441,24312,48648,92721,
%T A212388 170811,311886,589590,1220979,2864973,7450852,20309628,55305706,
%U A212388 146505451,373452808,913836082,2150455648,4887179761,10794337952,23375638064,50219351232
%N A212388 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 8).
%C A212388 Lengths of descents are unrestricted.
%H A212388 Alois P. Heinz, <a href="/A212388/b212388.txt">Table of n, a(n) for n = 0..750</a>
%H A212388 Vaclav Kotesovec, <a href="http://oeis.org/A212382/a212382.pdf">Asymptotic of subsequences of A212382</a>
%F A212388 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^8).
%F A212388 a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where p = 8 and r = 0.4098875088359862102..., s = 1.880071788712472133... are roots of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - _Vaclav Kotesovec_, Jul 16 2014
%e A212388 a(0) = 1: the empty path.
%e A212388 a(1) = 1: UD.
%e A212388 a(9) = 2: UDUDUDUDUDUDUDUDUD, UUUUUUUUUDDDDDDDDD.
%e A212388 a(10) = 11: UDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUDDDDDDDDD, UUUUUUUUUDDDDDDDDDUD, UUUUUUUUUDDDDDDDDUDD, UUUUUUUUUDDDDDDDUDDD, UUUUUUUUUDDDDDDUDDDD, UUUUUUUUUDDDDDUDDDDD, UUUUUUUUUDDDDUDDDDDD, UUUUUUUUUDDDUDDDDDDD, UUUUUUUUUDDUDDDDDDDD, UUUUUUUUUDUDDDDDDDDD.
%p A212388 b:= proc(x, y, u) option remember;
%p A212388       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212388       `if`(u, add (b(x-(8*t+1), y, false), t=0..(x-1)/8), 0)))
%p A212388     end:
%p A212388 a:= n-> b(n$2, true):
%p A212388 seq(a(n), n=0..40);
%p A212388 # second Maple program
%p A212388 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^8), A), x, n+1), x, n):
%p A212388 seq(a(n), n=0..40);
%Y A212388 Column k=8 of A212382.
%K A212388 nonn
%O A212388 0,10
%A A212388 _Alois P. Heinz_, May 12 2012