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

Author

Alois P. Heinz, Jun 22 2017

Keywords

• nonn

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.

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Wikipedia, Counting lattice paths

Crossrefs

Column k=1 of A288972.

Cf. A000217, A060747, A281874, A287846.

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)*
       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 *)