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A212364 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 5). 2
1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 23, 35, 57, 96, 161, 264, 425, 682, 1106, 1821, 3030, 5055, 8412, 13956, 23145, 38487, 64261, 107673, 180762, 303651, 510187, 857692, 1443597, 2433495, 4108299, 6943862, 11746362, 19883655, 33681015, 57096874, 96874214 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

LINKS

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

FORMULA

G.f. satisfies: A(x) = 1+A(x)*(x-x^5*(1-A(x))).

a(n) = a(n-1) + Sum_{k=1..n-5} a(k)*a(n-5-k) if n>0; a(0) = 1.

Recurrence: (n+5)*a(n) = (2*n+7)*a(n-1) - (n+2)*a(n-2) + (2*n-5)*a(n-5) + 2*(n-4)*a(n-6) - (n-10)*a(n-10). - Vaclav Kotesovec, Mar 20 2014

a(n) = Sum_{k=0..(n-1)/4} C(n-4*k,k)*C(n-4*k,k+1)/(n-4*k) for n>0, a(0)=1. - Vladimir Kruchinin, Jan 21 2019

EXAMPLE

a(0) = 1: the empty path.

a(1) = 1: UD.

a(5) = 1: UDUDUDUDUD.

a(6) = 2: UDUDUDUDUDUD, UUUUUUDDDDDD.

a(7) = 4: UDUDUDUDUDUDUD, UDUUUUUUDDDDDD, UUUUUUDDDDDDUD, UUUUUUDUDDDDDD.

a(8) = 7: UDUDUDUDUDUDUDUD, UDUDUUUUUUDDDDDD, UDUUUUUUDDDDDDUD, UDUUUUUUDUDDDDDD, UUUUUUDDDDDDUDUD, UUUUUUDUDDDDDDUD, UUUUUUDUDUDDDDDD.

MAPLE

a:= proc(n) option remember;

      `if`(n=0, 1, a(n-1) +add(a(k)*a(n-5-k), k=1..n-5))

    end:

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

# second Maple program:

a:= n-> coeff(series(RootOf(A=1+A*(x-x^5*(1-A)), A), x, n+1), x, n):

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

MATHEMATICA

CoefficientList[Series[(1-x+x^5-Sqrt[-4*x^5+(1-x+x^5)^2])/(2*x^5), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

CROSSREFS

Column k=5 of A212363.

Cf. A023432 (m=3), A023427 (m=4), this sequence (m=5), A212386(m=6).

Sequence in context: A317910 A065095 A005253 * A320591 A129339 A196719

Adjacent sequences:  A212361 A212362 A212363 * A212365 A212366 A212367

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 10 2012

STATUS

approved

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Last modified October 23 16:46 EDT 2019. Contains 328373 sequences. (Running on oeis4.)