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A212369 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 10). 2
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 68, 85, 112, 156, 226, 333, 490, 712, 1016, 1421, 1949, 2630, 3512, 4676, 6256, 8464, 11620, 16187, 22811, 32366, 46005, 65225, 91967, 128786, 179140, 247861, 341885, 471332, 651041, 902679 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,12

LINKS

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

FORMULA

G.f. satisfies: A(x) = 1+A(x)*(x-x^10*(1-A(x))).

a(n) = a(n-1) + Sum_{k=1..n-10} a(k)*a(n-10-k) if n>0; a(0) = 1.

EXAMPLE

a(0) = 1: the empty path.

a(1) = 1: UD.

a(11) = 2: UDUDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUUDDDDDDDDDDD.

a(12) = 4: UDUDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUUDDDDDDDDDDD, UUUUUUUUUUUDDDDDDDDDDDUD, UUUUUUUUUUUDUDDDDDDDDDDD.

MAPLE

a:= proc(n) option remember;

      `if`(n=0, 1, a(n-1) +add(a(k)*a(n-10-k), k=1..n-10))

    end:

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

# second Maple program:

a:= n-> coeff(series(RootOf(A=1+A*(x-x^10*(1-A)), A), x, n+1), x, n):

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

MATHEMATICA

With[{k = 10}, CoefficientList[Series[(1 - x + x^k - Sqrt[(1 - x + x^k)^2 - 4*x^k]) / (2*x^k), {x, 0, 50}], x]] (* Vaclav Kotesovec, Sep 02 2014 *)

CROSSREFS

Column k=10 of A212363.

Sequence in context: A025739 A000124 A152947 * A212368 A217838 A212367

Adjacent sequences:  A212366 A212367 A212368 * A212370 A212371 A212372

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 10 2012

STATUS

approved

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