|
|
A212367
|
|
Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 8).
|
|
2
|
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 47, 62, 87, 129, 197, 302, 457, 677, 980, 1392, 1957, 2752, 3907, 5630, 8237, 12187, 18123, 26927, 39810, 58472, 85381, 124234, 180677, 263375, 385538, 567036, 837306, 1239408, 1835867, 2717386, 4016173
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,10
|
|
LINKS
|
|
|
FORMULA
|
G.f. satisfies: A(x) = 1+A(x)*(x-x^8*(1-A(x))).
a(n) = a(n-1) + Sum_{k=1..n-8} a(k)*a(n-8-k) if n>0; a(0) = 1.
|
|
EXAMPLE
|
a(0) = 1: the empty path.
a(1) = 1: UD.
a(9) = 2: UDUDUDUDUDUDUDUDUD, UUUUUUUUUDDDDDDDDD.
a(10) = 4: UDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUDDDDDDDDD, UUUUUUUUUDDDDDDDDDUD, UUUUUUUUUDUDDDDDDDDD.
|
|
MAPLE
|
a:= proc(n) option remember;
`if`(n=0, 1, a(n-1) +add(a(k)*a(n-8-k), k=1..n-8))
end:
seq(a(n), n=0..60);
# second Maple program:
a:= n-> coeff(series(RootOf(A=1+A*(x-x^8*(1-A)), A), x, n+1), x, n):
seq(a(n), n=0..60);
|
|
MATHEMATICA
|
With[{k = 8}, 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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|