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A212366 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 7). 2
1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 22, 29, 38, 52, 76, 117, 184, 288, 442, 662, 972, 1414, 2063, 3047, 4572, 6952, 10645, 16303, 24857, 37672, 56821, 85541, 128948, 195103, 296548, 452501, 692053, 1058990, 1619311, 2473171, 3773889, 5757885, 8791090 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

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

FORMULA

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

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

EXAMPLE

a(0) = 1: the empty path.

a(1) = 1: UD.

a(8) = 2: UDUDUDUDUDUDUDUD, UUUUUUUUDDDDDDDD.

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

MAPLE

a:= proc(n) option remember;

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

    end:

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

# second Maple program:

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

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

MATHEMATICA

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 *)

CROSSREFS

Column k=7 of A212363.

Sequence in context: A225088 A175777 A098574 * A309838 A334251 A175776

Adjacent sequences:  A212363 A212364 A212365 * A212367 A212368 A212369

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 10 2012

STATUS

approved

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Last modified September 29 06:29 EDT 2020. Contains 337425 sequences. (Running on oeis4.)