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A212389 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 9).

%I A212389 #13 May 11 2015 19:48:41
%S A212389 1,1,1,1,1,1,1,1,1,1,2,12,67,287,1002,3004,8009,19449,43759,92380,
%T A212389 184787,353137,650497,1170632,2110021,3977161,8271836,19536661,
%U A212389 51111062,140210129,385123916,1032218316,2670065961,6645249777,15922990909,36823807747,82485177457
%N A212389 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 9).
%C A212389 Lengths of descents are unrestricted.
%H A212389 Alois P. Heinz, <a href="/A212389/b212389.txt">Table of n, a(n) for n = 0..800</a>
%H A212389 Vaclav Kotesovec, <a href="http://oeis.org/A212382/a212382.pdf">Asymptotic of subsequences of A212382</a>
%F A212389 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^9).
%F A212389 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 = 9 and r = 0.4164039515514120671..., s = 1.882616423435763466... 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 A212389 a(0) = 1: the empty path.
%e A212389 a(1) = 1: UD.
%e A212389 a(10) = 2: UDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUDDDDDDDDDD.
%e A212389 a(11) = 12: UDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUDDDDDDDDDD, UUUUUUUUUUDDDDDDDDDDUD, UUUUUUUUUUDDDDDDDDDUDD, UUUUUUUUUUDDDDDDDDUDDD, UUUUUUUUUUDDDDDDDUDDDD, UUUUUUUUUUDDDDDDUDDDDD, UUUUUUUUUUDDDDDUDDDDDD, UUUUUUUUUUDDDDUDDDDDDD, UUUUUUUUUUDDDUDDDDDDDD, UUUUUUUUUUDDUDDDDDDDDD, UUUUUUUUUUDUDDDDDDDDDD.
%p A212389 b:= proc(x, y, u) option remember;
%p A212389       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212389       `if`(u, add (b(x-(9*t+1), y, false), t=0..(x-1)/9), 0)))
%p A212389     end:
%p A212389 a:= n-> b(n$2, true):
%p A212389 seq(a(n), n=0..40);
%p A212389 # second Maple program:
%p A212389 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^9), A), x, n+1), x, n):
%p A212389 seq(a(n), n=0..40);
%Y A212389 Column k=9 of A212382.
%K A212389 nonn
%O A212389 0,11
%A A212389 _Alois P. Heinz_, May 12 2012