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A287846
Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has exactly one peak.
9
1, 1, 0, 2, 0, 4, 6, 8, 24, 52, 96, 212, 504, 1072, 2352, 5288, 11928, 26800, 60336, 136304, 308928, 701248, 1593120, 3622016, 8245008, 18787360, 42836928, 97724384, 223052784, 509338816, 1163512032, 2658731648, 6077117376, 13893874624, 31771515648
OFFSET
0,4
COMMENTS
All terms with n > 1 are even.
LINKS
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 15.
EXAMPLE
. a(1) = 1: /\ .
.
. a(3) = 2: /\ /\
. /\/ \ / \/\ .
.
. a(5) = 4:
. /\ /\ /\ /\
. /\/ \ / \/\ /\/ \ / \/\
. /\/ \ /\/ \ / \/\ / \/\ .
MAPLE
b:= proc(n, j) option remember; `if`(n=j or n=0, 1, add(
b(n-j, i)*binomial(j-1, i-2)*i, i=1..min(j+2, n-j)))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..35);
MATHEMATICA
b[n_, j_] := b[n, j] = If[n == j || n == 0, 1, Sum[b[n - j, i]*Binomial[j - 1, i - 2]*i, {i, 1, Min[j + 2, n - j]}]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 23 2018, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 01 2017
STATUS
approved