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A288141 Number of Dyck paths of semilength n such that the number of peaks is strongly decreasing from lower to higher levels. 4
1, 1, 1, 1, 4, 5, 10, 22, 46, 148, 324, 722, 1843, 4634, 12537, 34248, 95711, 266761, 724689, 1983267, 5553902, 15900083, 46201546, 135511171, 400668869, 1189723253, 3535186203, 10516298421, 31405658622, 94378367065, 285623516777, 870481565252, 2671088133010 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

Wikipedia, Counting lattice paths

EXAMPLE

a(5) = 5:

                     /\        /\        /\        /\

  /\/\/\/\/\  /\/\/\/  \  /\/\/  \/\  /\/  \/\/\  /  \/\/\/\

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(k+1, i-j)..min(n-j, i-1)), i=1..n-j))

    end:

a:= n-> `if`(n=0, 1, add(b(n, k$2), k=1..n)):

seq(a(n), n=0..34);

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, Max[k + 1, i - j], Min[n - j, i - 1]}], {i, 1, n - j}]];

a[n_] := If[n == 0, 1, Sum[b[n, k, k], {k, 1, n}]];

Table[a[n], {n, 0, 34}] (* Jean-Fran├žois Alcover, May 29 2018, from Maple *)

CROSSREFS

Cf. A000108, A008930, A048285, A288140, A288146, A288147.

Sequence in context: A049898 A166577 A242960 * A203853 A109675 A052508

Adjacent sequences:  A288138 A288139 A288140 * A288142 A288143 A288144

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 05 2017

STATUS

approved

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Last modified May 25 11:07 EDT 2020. Contains 334592 sequences. (Running on oeis4.)