A289030 Number of Dyck paths having exactly n peaks in each of the levels 1,2,3 and no other peaks.

1, 10, 471, 27076, 1713955, 114751470, 7969151855, 567878871304, 41247976697019, 3040572724077010, 226777538499783271, 17076122335343354700, 1296037531424347164115, 99025149551454886937590, 7609414766853344476768095, 587623058661705739915402256

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 [3*n+2, 6*n] for n>0.

Examples

			. a(1) = 10:
.
.        /\        /\          /\        /\
.     /\/  \      /  \/\    /\/  \      /  \/\
.  /\/      \  /\/      \  /      \/\  /      \/\
.
.                /\        /\                /\
.           /\  /  \      /  \  /\    /\    /  \
.        /\/  \/    \  /\/    \/  \  /  \/\/    \
.
.              /\        /\            /\
.         /\  /  \      /  \    /\    /  \  /\
.        /  \/    \/\  /    \/\/  \  /    \/  \/\  .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..523
• Wikipedia, Counting lattice paths

Crossrefs

Row n=3 of A288972.

Programs

Maple

b:= proc(n, k, j, v) option remember; `if`(n=j, `if`(v=1, 1, 0),
      `if`(v<2, 0, add(b(n-j, k, i, v-1)*(binomial(i, k)*
       binomial(j-1, i-1-k)), i=1..min(j+k, n-j))))
    end:
a:= n-> `if`(n=0, 1, add(b(w, n$2, 3), w=3*n+2..6*n)):
seq(a(n), n=0..15);

Mathematica

b[n_, k_, j_, v_]:=b[n, k, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, k, i, v - 1] Binomial[i, k] Binomial[j - 1, i - 1 - k], {i, Min[j + k, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[w, n, n, 3], {w, 3n + 2, 6n}]]; Table[a[n], {n, 0, 15}] (* Indranil Ghosh, Jul 06 2017, after maple code *)