A212366 - OEIS

%I #10 Jun 23 2017 08:00:39

%S 1,1,1,1,1,1,1,1,2,4,7,11,16,22,29,38,52,76,117,184,288,442,662,972,

%T 1414,2063,3047,4572,6952,10645,16303,24857,37672,56821,85541,128948,

%U 195103,296548,452501,692053,1058990,1619311,2473171,3773889,5757885,8791090

%N Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 7).

%H Alois P. Heinz, <a href="/A212366/b212366.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. satisfies: A(x) = 1+A(x)*(x-x^7*(1-A(x))).

%F a(n) = a(n-1) + Sum_{k=1..n-7} a(k)*a(n-7-k) if n>0; a(0) = 1.

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

%e a(1) = 1: UD.

%e a(8) = 2: UDUDUDUDUDUDUDUD, UUUUUUUUDDDDDDDD.

%e a(9) = 4: UDUDUDUDUDUDUDUDUD, UDUUUUUUUUDDDDDDDD, UUUUUUUUDDDDDDDDUD, UUUUUUUUDUDDDDDDDD.

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

%p       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-7-k), k=1..n-7))

%p     end:

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

%p # second Maple program:

%p a:= n-> coeff(series(RootOf(A=1+A*(x-x^7*(1-A)), A), x, n+1), x, n):

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

%t With[{k = 7}, 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=7 of A212363.

%K nonn

%O 0,9

%A _Alois P. Heinz_, May 10 2012