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.
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
Keywords
Examples
a(4) = 6:
/\ /\ /\ /\/\ /\/\
/\/\/\/\ /\/\/ \ /\/ \/\ / \/\/\ /\/ \ / \/\
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..70
- Wikipedia, Counting lattice paths
Programs
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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 *)