A288940 Number of Dyck paths having n (positive) levels and exactly n peaks per level.

1, 1, 9, 27076, 147556480375, 4711342006036190504484, 2162932174406679548553402518043252929, 29605698225102450501737027784037791564430800582087459328, 22346336234943531646124131709622442581521043809236751640919325993842966011809319

Offset: 0

Author

Alois P. Heinz, Jun 19 2017

Keywords

• nonn

Comments

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

Examples

			. a(1) = 1:       /\  .
.
. a(2) = 9:           /\/\        /\/\        /\/\             /\  /\
.                /\/\/    \    /\/    \/\    /    \/\/\   /\/\/  \/  \
.
.    /\    /\      /\  /\      /\      /\    /\    /\      /\  /\
. /\/  \/\/  \  /\/  \/  \/\  /  \/\/\/  \  /  \/\/  \/\  /  \/  \/\/\ .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..15
• Wikipedia, Counting lattice paths

Crossrefs

Main diagonal of A288972.

Cf. A288318.

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(k, n$3), k=n^2+n-1..n^2*(n+1)/2)):
seq(a(n), n=0..7);

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[k, n, n, n], {k, n^2 + n - 1, n^2*(n + 1)/2}]]; Table[a[n], {n, 0, 8}] (* Indranil Ghosh, Jul 06 2017, after Maple code *)