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.

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

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-François 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