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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jun 02 2018, from Maple *)

CROSSREFS

Cf. A000108, A287993.

Sequence in context: A077621 A298736 A324617 * A081394 A184387 A295185

Adjacent sequences:  A287986 A287987 A287988 * A287990 A287991 A287992

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 04 2017

STATUS

approved

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Last modified February 29 08:26 EST 2020. Contains 332355 sequences. (Running on oeis4.)