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A212368 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 9). 2
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 57, 73, 99, 142, 211, 317, 473, 694, 997, 1402, 1937, 2648, 3614, 4967, 6917, 9782, 14023, 20284, 29438, 42647, 61457, 87963, 125093, 177074, 250157, 353692, 501658, 714768, 1023296, 1470843 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

LINKS

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

FORMULA

G.f. satisfies: A(x) = 1+A(x)*(x-x^9*(1-A(x))).

a(n) = a(n-1) + Sum_{k=1..n-9} a(k)*a(n-9-k) if n>0; a(0) = 1.

EXAMPLE

a(0) = 1: the empty path.

a(1) = 1: UD.

a(10) = 2: UDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUDDDDDDDDDD.

a(11) = 4: UDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUDDDDDDDDDD, UUUUUUUUUUDDDDDDDDDDUD, UUUUUUUUUUDUDDDDDDDDDD.

MAPLE

a:= proc(n) option remember;

      `if`(n=0, 1, a(n-1) +add(a(k)*a(n-9-k), k=1..n-9))

    end:

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

# second Maple program:

a:= n-> coeff(series(RootOf(A=1+A*(x-x^9*(1-A)), A), x, n+1), x, n):

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

MATHEMATICA

With[{k = 9}, 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=9 of A212363.

Sequence in context: A000124 A152947 A212369 * A217838 A212367 A225088

Adjacent sequences:  A212365 A212366 A212367 * A212369 A212370 A212371

KEYWORD

nonn,changed

AUTHOR

Alois P. Heinz, May 10 2012

STATUS

approved

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Last modified June 28 16:56 EDT 2017. Contains 288839 sequences.