A289020 Number of Dyck paths having exactly one peak in each of the levels 1,...,n and no other peaks.

1, 1, 2, 10, 92, 1348, 28808, 845800, 32664944, 1605553552, 97868465696, 7245440815264, 640359291096512, 66598657958731840, 8051483595083729024, 1119653568781387712128, 177465810459239319017216, 31804047327185301634148608, 6398867435594240638421950976

Offset

0,3

Comments

The semilengths of Dyck paths counted by a(n) are elements of the integer interval [2*n-1, n*(n+1)/2] = [A060747(n), A000217(n)] for n>0.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Wikipedia, Counting lattice paths

Example

. a(2) = 2:      /\    /\
.             /\/  \  /  \/\  .

Maple

b:= proc(n, j, v) option remember; `if`(n=j,
      `if`(v=1, 1, 0), `if`(v<2, 0, add(b(n-j, i, v-1)*
       i*binomial(j-1, i-2), i=1..min(j+1, n-j))))
    end:
a:= n-> `if`(n=0, 1, add(b(w, 1, n), w=2*n-1..n*(n+1)/2)):
seq(a(n), n=0..18);

Mathematica

b[n_, j_, v_]:=b[n, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, i, v - 1]*i*Binomial[j - 1, i - 2], {i, Min[j + 1, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[w, 1, n], {w, 2*n - 1, n*(n + 1)/2}]]; Table[a[n], {n, 0, 18}] (* Indranil Ghosh, Jul 06 2017, after Maple code *)

Crossrefs

Column k=1 of A288972.

Cf. A000217, A060747, A281874, A287846.

Sequence in context: A182952 A108209 A111773 * A195415 A336271 A181084

Adjacent sequences: A289017 A289018 A289019 * A289021 A289022 A289023

Keyword

nonn

Author

Alois P. Heinz, Jun 22 2017

Status

approved