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%I #10 Jun 23 2017 07:55:20
%S 1,1,1,1,1,1,1,1,1,1,2,4,7,11,16,22,29,37,46,57,73,99,142,211,317,473,
%T 694,997,1402,1937,2648,3614,4967,6917,9782,14023,20284,29438,42647,
%U 61457,87963,125093,177074,250157,353692,501658,714768,1023296,1470843
%N Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 9).
%H Alois P. Heinz, <a href="/A212368/b212368.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f. satisfies: A(x) = 1+A(x)*(x-x^9*(1-A(x))).
%F a(n) = a(n-1) + Sum_{k=1..n-9} a(k)*a(n-9-k) if n>0; a(0) = 1.
%e a(0) = 1: the empty path.
%e a(1) = 1: UD.
%e a(10) = 2: UDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUDDDDDDDDDD.
%e a(11) = 4: UDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUDDDDDDDDDD, UUUUUUUUUUDDDDDDDDDDUD, UUUUUUUUUUDUDDDDDDDDDD.
%p a:= proc(n) option remember;
%p `if`(n=0, 1, a(n-1) +add(a(k)*a(n-9-k), k=1..n-9))
%p end:
%p seq(a(n), n=0..60);
%p # second Maple program:
%p a:= n-> coeff(series(RootOf(A=1+A*(x-x^9*(1-A)), A), x, n+1), x, n):
%p seq(a(n), n=0..60);
%t With[{k = 9}, CoefficientList[Series[(1 - x + x^k - Sqrt[(1 - x + x^k)^2 - 4*x^k]) / (2*x^k), {x, 0, 40}], x]] (* _Vaclav Kotesovec_, Sep 02 2014 *)
%Y Column k=9 of A212363.
%K nonn
%O 0,11
%A _Alois P. Heinz_, May 10 2012
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