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A287963
Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has one or two peaks.
4
1, 1, 1, 2, 5, 10, 28, 71, 194, 532, 1495, 4256, 12176, 35251, 102664, 300260, 881909, 2599948, 7688164, 22788527, 67676144, 201308938, 599676445, 1788564038, 5339905904, 15956230705, 47713265536, 142763240666, 427390085963, 1280058256294, 3835332884686
OFFSET
0,4
LINKS
EXAMPLE
. a(3) = 2: /\ /\
. /\/ \ / \/\ .
.
. a(4) = 5: /\ /\ /\/\ /\ /\/\
. /\/\/ \ /\/ \/\ /\/ \ / \/\/\ / \/\ .
MAPLE
b:= proc(n, j) option remember; `if`(n=j, 1, add(
b(n-j, i)*i*(binomial(j-1, i-2) +(i-1)/2*
binomial(j-1, i-3)), i=2..min(j+3, n-j)))
end:
a:= n-> `if`(n=0, 1, b(n, 1)+b(n, 2)):
seq(a(n), n=0..35);
MATHEMATICA
b[n_, j_] := b[n, j] = If[n == j, 1, Sum[b[n - j, i]*i*(Binomial[j - 1, i - 2] + (i - 1)/2*Binomial[j - 1, i - 3]), {i, 2, Min[j + 3, n - j]}]];
a[n_] := If[n == 0, 1, b[n, 1] + b[n, 2]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 29 2018, from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 03 2017
STATUS
approved