cp's OEIS Frontend

A212367 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 8).

%I A212367 #11 Jun 23 2017 07:53:15
%S A212367 1,1,1,1,1,1,1,1,1,2,4,7,11,16,22,29,37,47,62,87,129,197,302,457,677,
%T A212367 980,1392,1957,2752,3907,5630,8237,12187,18123,26927,39810,58472,
%U A212367 85381,124234,180677,263375,385538,567036,837306,1239408,1835867,2717386,4016173
%N A212367 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 8).
%H A212367 Alois P. Heinz, <a href="/A212367/b212367.txt">Table of n, a(n) for n = 0..1000</a>
%F A212367 G.f. satisfies: A(x) = 1+A(x)*(x-x^8*(1-A(x))).
%F A212367 a(n) = a(n-1) + Sum_{k=1..n-8} a(k)*a(n-8-k) if n>0; a(0) = 1.
%e A212367 a(0) = 1: the empty path.
%e A212367 a(1) = 1: UD.
%e A212367 a(9) = 2: UDUDUDUDUDUDUDUDUD, UUUUUUUUUDDDDDDDDD.
%e A212367 a(10) = 4: UDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUDDDDDDDDD, UUUUUUUUUDDDDDDDDDUD, UUUUUUUUUDUDDDDDDDDD.
%p A212367 a:= proc(n) option remember;
%p A212367       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-8-k), k=1..n-8))
%p A212367     end:
%p A212367 seq(a(n), n=0..60);
%p A212367 # second Maple program:
%p A212367 a:= n-> coeff(series(RootOf(A=1+A*(x-x^8*(1-A)), A), x, n+1), x, n):
%p A212367 seq(a(n), n=0..60);
%t A212367 With[{k = 8}, 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 A212367 Column k=8 of A212363.
%K A212367 nonn
%O A212367 0,10
%A A212367 _Alois P. Heinz_, May 10 2012