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A212369 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 10).

%I A212369 #13 Dec 20 2014 23:03:40
%S A212369 1,1,1,1,1,1,1,1,1,1,1,2,4,7,11,16,22,29,37,46,56,68,85,112,156,226,
%T A212369 333,490,712,1016,1421,1949,2630,3512,4676,6256,8464,11620,16187,
%U A212369 22811,32366,46005,65225,91967,128786,179140,247861,341885,471332,651041,902679
%N A212369 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 10).
%H A212369 Alois P. Heinz, <a href="/A212369/b212369.txt">Table of n, a(n) for n = 0..1000</a>
%F A212369 G.f. satisfies: A(x) = 1+A(x)*(x-x^10*(1-A(x))).
%F A212369 a(n) = a(n-1) + Sum_{k=1..n-10} a(k)*a(n-10-k) if n>0; a(0) = 1.
%e A212369 a(0) = 1: the empty path.
%e A212369 a(1) = 1: UD.
%e A212369 a(11) = 2: UDUDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUUDDDDDDDDDDD.
%e A212369 a(12) = 4: UDUDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUUDDDDDDDDDDD, UUUUUUUUUUUDDDDDDDDDDDUD, UUUUUUUUUUUDUDDDDDDDDDDD.
%p A212369 a:= proc(n) option remember;
%p A212369       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-10-k), k=1..n-10))
%p A212369     end:
%p A212369 seq(a(n), n=0..60);
%p A212369 # second Maple program:
%p A212369 a:= n-> coeff(series(RootOf(A=1+A*(x-x^10*(1-A)), A), x, n+1), x, n):
%p A212369 seq(a(n), n=0..60);
%t A212369 With[{k = 10}, CoefficientList[Series[(1 - x + x^k - Sqrt[(1 - x + x^k)^2 - 4*x^k]) / (2*x^k), {x, 0, 50}], x]] (* _Vaclav Kotesovec_, Sep 02 2014 *)
%Y A212369 Column k=10 of A212363.
%K A212369 nonn
%O A212369 0,12
%A A212369 _Alois P. Heinz_, May 10 2012