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

Author

Alois P. Heinz, Jun 23 2017

Keywords

• nonn

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.

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..60
• Wikipedia, Counting lattice paths

Crossrefs

Column k=2 of A288972.

Programs

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 *)