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A288147 Number of Dyck paths of semilength n such that the number of peaks is strongly increasing from lower to higher levels and no positive level is peakless. 4
1, 1, 1, 1, 3, 6, 12, 31, 68, 186, 506, 1299, 3481, 9712, 27692, 79587, 232743, 694896, 2086245, 6248158, 18771510, 57007483, 175149700, 542313513, 1688360997, 5288335561, 16679137617, 52933231538, 168768966207, 539981776609, 1733555552587, 5587076558809 (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) = 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(k-1, 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[1, i - j], Min[k - 1, 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, A288141, A288146.

Sequence in context: A049941 A219634 A252696 * A026079 A066710 A033648

Adjacent sequences:  A288144 A288145 A288146 * A288148 A288149 A288150

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 05 2017

STATUS

approved

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Last modified October 16 00:29 EDT 2021. Contains 348034 sequences. (Running on oeis4.)