A287989 Number of Dyck paths of semilength n such that all positive levels up to the highest level have a positive number of peaks and the number of peaks of adjacent levels is different.

1, 1, 1, 1, 6, 10, 27, 84, 226, 770, 2390, 7579, 25222, 84299, 285284, 976105, 3386494, 11858759, 41782516, 148205047, 529101609, 1899680494, 6854597493, 24847293152, 90460431604, 330654288724, 1213033321450, 4465027739962, 16486012746085, 61044028354833

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..150

Wikipedia, Counting lattice paths

Example

. (4) = 6:
.                    /\      /\      /\          /\/\    /\/\
.     /\/\/\/\  /\/\/  \  /\/  \/\  /  \/\/\  /\/    \  /    \/\  .

Maple

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(add(
       b(n-j, t, i)*binomial(i, t)*binomial(j-1, i-1-t),
       t={$max(1, i-j)..min(n-j, i-1)} minus {k}), i=1..n-j))
    end:
a:= n-> `if`(n=0, 1, add(b(n, k$2), k=1..n)):
seq(a(n), n=0..30);

Mathematica

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Sum[b[n - j, t, i]* Binomial[i, t]*Binomial[j - 1, i - 1 - t], {t, Range[Max[1, i - j], Min[n - j, i - 1]] ~Complement~ {k}}], {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, Jun 02 2018, from Maple *)

Crossrefs

Cf. A000108, A287993.

Sequence in context: A336845 A324617 A359145 * A081394 A184387 A295185

Adjacent sequences: A287986 A287987 A287988 * A287990 A287991 A287992

Keyword

nonn

Author

Alois P. Heinz, Jun 04 2017

Status

approved