login
A212364
Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 5).
10
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
OFFSET
0,7
LINKS
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 10 2012
STATUS
approved