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A212366 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 7).

%I A212366 #10 Jun 23 2017 08:00:39
%S A212366 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 A212366 1414,2063,3047,4572,6952,10645,16303,24857,37672,56821,85541,128948,
%U A212366 195103,296548,452501,692053,1058990,1619311,2473171,3773889,5757885,8791090
%N A212366 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 7).
%H A212366 Alois P. Heinz, <a href="/A212366/b212366.txt">Table of n, a(n) for n = 0..1000</a>
%F A212366 G.f. satisfies: A(x) = 1+A(x)*(x-x^7*(1-A(x))).
%F A212366 a(n) = a(n-1) + Sum_{k=1..n-7} a(k)*a(n-7-k) if n>0; a(0) = 1.
%e A212366 a(0) = 1: the empty path.
%e A212366 a(1) = 1: UD.
%e A212366 a(8) = 2: UDUDUDUDUDUDUDUD, UUUUUUUUDDDDDDDD.
%e A212366 a(9) = 4: UDUDUDUDUDUDUDUDUD, UDUUUUUUUUDDDDDDDD, UUUUUUUUDDDDDDDDUD, UUUUUUUUDUDDDDDDDD.
%p A212366 a:= proc(n) option remember;
%p A212366       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-7-k), k=1..n-7))
%p A212366     end:
%p A212366 seq(a(n), n=0..50);
%p A212366 # second Maple program:
%p A212366 a:= n-> coeff(series(RootOf(A=1+A*(x-x^7*(1-A)), A), x, n+1), x, n):
%p A212366 seq(a(n), n=0..50);
%t A212366 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 A212366 Column k=7 of A212363.
%K A212366 nonn
%O A212366 0,9
%A A212366 _Alois P. Heinz_, May 10 2012