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A287993
Number of Dyck paths of semilength n such that all positive levels up to the highest level have a positive number of peaks and all the level peak numbers are distinct.
3
1, 1, 1, 1, 6, 10, 21, 52, 147, 564, 1651, 4440, 12499, 36853, 116476, 390774, 1352215, 4593736, 15057127, 48419013, 156073723, 511324062, 1713185811, 5878350249, 20574046540, 72771206715, 257475113013, 905430711156, 3160767910928, 10981916671027
OFFSET
0,5
LINKS
EXAMPLE
a(4) = 6:
/\ /\ /\ /\/\ /\/\
/\/\/\/\ /\/\/ \ /\/ \/\ / \/\/\ /\/ \ / \/\
MAPLE
b:= proc(n, s, j) option remember; `if`(n=j, 1, add(add(
b(n-j, s union {t}, i)*binomial(i, t)*binomial(j-1, i-1-t),
t={$max(1, i-j)..min(n-j, i-1)} minus s), i=1..n-j))
end:
a:= n-> `if`(n=0, 1, add(b(n, {k}, k), k=1..n)):
seq(a(n), n=0..30);
MATHEMATICA
b[n_, s_, j_] := b[n, s, j] = If[n==j, 1, Sum[Sum[b[n-j, s ~Union~ {t}, i]* Binomial[i, t]*Binomial[j-1, i-1-t], {t, Range[Max[1, i - j], Min[n - j, i - 1]] ~Complement~ s}], {i, 1, n - j}]];
a[n_] := If[n == 0, 1, Sum[b[n, {k}, k], {k, 1, n}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 31 2018, from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 04 2017
STATUS
approved