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A212367 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 8). 2
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, 980, 1392, 1957, 2752, 3907, 5630, 8237, 12187, 18123, 26927, 39810, 58472, 85381, 124234, 180677, 263375, 385538, 567036, 837306, 1239408, 1835867, 2717386, 4016173 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f. satisfies: A(x) = 1+A(x)*(x-x^8*(1-A(x))).

a(n) = a(n-1) + Sum_{k=1..n-8} a(k)*a(n-8-k) if n>0; a(0) = 1.

EXAMPLE

a(0) = 1: the empty path.

a(1) = 1: UD.

a(9) = 2: UDUDUDUDUDUDUDUDUD, UUUUUUUUUDDDDDDDDD.

a(10) = 4: UDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUDDDDDDDDD, UUUUUUUUUDDDDDDDDDUD, UUUUUUUUUDUDDDDDDDDD.

MAPLE

a:= proc(n) option remember;

      `if`(n=0, 1, a(n-1) +add(a(k)*a(n-8-k), k=1..n-8))

    end:

seq(a(n), n=0..60);

# second Maple program:

a:= n-> coeff(series(RootOf(A=1+A*(x-x^8*(1-A)), A), x, n+1), x, n):

seq(a(n), n=0..60);

MATHEMATICA

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 *)

CROSSREFS

Column k=8 of A212363.

Sequence in context: A212369 A212368 A217838 * A225088 A175777 A098574

Adjacent sequences:  A212364 A212365 A212366 * A212368 A212369 A212370

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 10 2012

STATUS

approved

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Last modified August 10 04:38 EDT 2020. Contains 336368 sequences. (Running on oeis4.)