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A212365 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 6).

%I A212365 #10 Jun 23 2017 08:00:07
%S A212365 1,1,1,1,1,1,1,2,4,7,11,16,22,30,43,66,106,172,275,429,656,996,1522,
%T A212365 2360,3714,5897,9376,14852,23410,36788,57828,91187,144413,229561,
%U A212365 365678,582766,928280,1477877,2353062,3749721,5983631,9562565,15300700,24501417
%N A212365 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 6).
%H A212365 Alois P. Heinz, <a href="/A212365/b212365.txt">Table of n, a(n) for n = 0..1000</a>
%F A212365 G.f. satisfies: A(x) = 1+A(x)*(x-x^6*(1-A(x))).
%F A212365 a(n) = a(n-1) + Sum_{k=1..n-6} a(k)*a(n-6-k) if n>0; a(0) = 1.
%e A212365 a(0) = 1: the empty path.
%e A212365 a(1) = 1: UD.
%e A212365 a(6) = 1: UDUDUDUDUDUD.
%e A212365 a(7) = 2: UDUDUDUDUDUDUD, UUUUUUUDDDDDDD.
%e A212365 a(8) = 4: UDUDUDUDUDUDUDUD, UDUUUUUUUDDDDDDD, UUUUUUUDDDDDDDUD, UUUUUUUDUDDDDDDD.
%p A212365 a:= proc(n) option remember;
%p A212365       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-6-k), k=1..n-6))
%p A212365     end:
%p A212365 seq(a(n), n=0..50);
%p A212365 # second Maple program:
%p A212365 a:= n-> coeff(series(RootOf(A=1+A*(x-x^6*(1-A)), A), x, n+1), x, n):
%p A212365 seq(a(n), n=0..50);
%t A212365 CoefficientList[Series[(1 - x + x^6 - Sqrt[(1-x+x^6)^2 - 4*x^6])/ (2*x^6), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Sep 02 2014 *)
%Y A212365 Column k=6 of A212363.
%K A212365 nonn
%O A212365 0,8
%A A212365 _Alois P. Heinz_, May 10 2012