A212365 - OEIS

%I #10 Jun 23 2017 08:00:07

%S 1,1,1,1,1,1,1,2,4,7,11,16,22,30,43,66,106,172,275,429,656,996,1522,

%T 2360,3714,5897,9376,14852,23410,36788,57828,91187,144413,229561,

%U 365678,582766,928280,1477877,2353062,3749721,5983631,9562565,15300700,24501417

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

%H Alois P. Heinz, <a href="/A212365/b212365.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. satisfies: A(x) = 1+A(x)*(x-x^6*(1-A(x))).

%F a(n) = a(n-1) + Sum_{k=1..n-6} a(k)*a(n-6-k) if n>0; a(0) = 1.

%e a(0) = 1: the empty path.

%e a(1) = 1: UD.

%e a(6) = 1: UDUDUDUDUDUD.

%e a(7) = 2: UDUDUDUDUDUDUD, UUUUUUUDDDDDDD.

%e a(8) = 4: UDUDUDUDUDUDUDUD, UDUUUUUUUDDDDDDD, UUUUUUUDDDDDDDUD, UUUUUUUDUDDDDDDD.

%p a:= proc(n) option remember;

%p       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-6-k), k=1..n-6))

%p     end:

%p seq(a(n), n=0..50);

%p # second Maple program:

%p a:= n-> coeff(series(RootOf(A=1+A*(x-x^6*(1-A)), A), x, n+1), x, n):

%p seq(a(n), n=0..50);

%t 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 Column k=6 of A212363.

%K nonn

%O 0,8

%A _Alois P. Heinz_, May 10 2012