A289054 Number of Dyck paths having exactly two peaks in each of the levels 1,...,n and no other peaks.

1, 1, 9, 471, 82899, 36913581, 34878248649, 62045165964951, 190543753640526939, 945931782247964900901, 7209377339218632463758129, 80920117567254715984058542191, 1292645840976784584918218615760819, 28557854803885245556927129118200208781

Offset

0,3

Comments

The semilengths of Dyck paths counted by a(n) are elements of the integer interval [3*n-1, n*(n+1)] for n>0.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..60

Wikipedia, Counting lattice paths

Example

. a(2) = 9:           /\/\        /\/\        /\/\             /\  /\
.                /\/\/    \    /\/    \/\    /    \/\/\   /\/\/  \/  \
.
.    /\    /\      /\  /\      /\      /\    /\    /\      /\  /\
. /\/  \/\/  \  /\/  \/  \/\  /  \/\/\/  \  /  \/\/  \/\  /  \/  \/\/\ .

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)*(binomial(i, 2)*
       binomial(j-1, i-3)), i=1..min(j+2, n-j))))
    end:
a:= n-> `if`(n=0, 1, add(b(w, 2, n), w=3*n-1..n*(n+1))):
seq(a(n), n=0..15);

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] Binomial[i, 2] Binomial[j - 1, i - 3], {i, Min[j + 2, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[w, 2, n], {w, 3*n - 1, n(n + 1)}]]; Table[a[n], {n, 0, 15}] (* Indranil Ghosh, Jul 06 2017, after Maple code *)

Crossrefs

Column k=2 of A288972.

Sequence in context: A265444 A285371 A061175 * A166879 A213447 A128947

Adjacent sequences: A289051 A289052 A289053 * A289055 A289056 A289057

Keyword

nonn

Author

Alois P. Heinz, Jun 23 2017

Status

approved