|
|
A334642
|
|
a(n) is the total number of down steps between the first and second up steps in all 2_1-Dyck paths of length 3*n. A 2_1-Dyck path is a lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.
|
|
5
|
|
|
0, 3, 9, 32, 139, 669, 3430, 18360, 101403, 573551, 3305445, 19340100, 114579348, 685962172, 4143459504, 25220816752, 154545611355, 952583230899, 5902090839715, 36738469359480, 229636903762035, 1440759023752125, 9070230371741490, 57278432955350880
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
For n = 1, there is no 2nd up step, a(1) = 3 enumerates the total number of down steps between the 1st up step and the end of the path.
|
|
LINKS
|
|
|
FORMULA
|
a(0) = 0 and a(n) = 2*binomial(3*n, n)/(n+1) - binomial(3*n+1, n)/(n+1) + 4*binomial(3*(n-1), n-1)/n - 2*[n=1] for n > 0, where [ ] is the Iverson bracket.
|
|
EXAMPLE
|
For n = 1, the 2_1-Dyck paths are UDD, DUD. This corresponds to a(1) = 2 + 1 = 3 down steps between the 1st up step and the end of the path.
For n = 2, the 2_1-Dyck paths are UUDDDD, UDUDDD, UDDUDD, UDDDUD, DUDDUD, DUDUDD, DUUDDD. In total, there are a(2) = 0 + 1 + 2 + 3 + 2 + 1 + 0 = 9 down steps between the 1st and 2nd up step.
|
|
MATHEMATICA
|
a[0] = 0; a[n_] := 2 * Binomial[3*n, n]/(n + 1) - Binomial[3*n + 1, n]/(n + 1) + 4 * Binomial[3*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 24, 0] (* Amiram Eldar, May 09 2020 *)
|
|
PROG
|
(PARI) a(n) = if (n==0, 0, 2*binomial(3*n, n)/(n+1) - binomial(3*n+1, n)/(n+1) + 4*binomial(3*(n-1), n-1)/n - 2*(n==1)); \\ Michel Marcus, May 09 2020
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|