|
|
A288111
|
|
Number of Dyck paths of semilength n such that each level has exactly four peaks or no peaks.
|
|
2
|
|
|
1, 0, 0, 0, 1, 1, 4, 7, 11, 22, 81, 235, 673, 2063, 5716, 13627, 33752, 95729, 298232, 946563, 2977953, 9147328, 27004159, 76880498, 217826819, 637089405, 1949908577, 6160707450, 19627448025, 61909478550, 191681762379, 583025396879, 1756696160636
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,7
|
|
LINKS
|
|
|
EXAMPLE
|
. a(6) = 4:
. /\/\/\/\
. /\ /\/\/\ /\/\ /\/\ /\/\/\ /\ / \
. / \/ \ / \/ \ / \/ \ / \ .
|
|
MAPLE
|
b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
*binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
end:
a:= n-> `if`(n=0, 1, b(n, 4$2)):
seq(a(n), n=0..40);
|
|
MATHEMATICA
|
b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(Binomial[j - 1, i - 1] + Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
a[n_] := If[n == 0, 1, b[n, 4, 4]];
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|