A288743 Number of Dyck paths of semilength n such that the maximal number of peaks per level equals two.

1, 1, 7, 18, 59, 193, 616, 1955, 6244, 19926, 63490, 202068, 642816, 2044571, 6502193, 20673020, 65714586, 208870774, 663868055, 2109997964, 6706282384, 21315049217, 67748772174, 215343287489, 684507346839, 2175916952697, 6917096914771, 21989855308501

Offset: 2

Author

Alois P. Heinz, Jun 14 2017

Keywords

• nonn

Examples

			. a(4) = 7:       /\      /\        /\/\    /\        /\  /\
.            /\/\/  \  /\/  \/\  /\/    \  /  \/\/\  /  \/  \ .
.
.                        /\/\
.             /\/\      /    \
.            /    \/\  /      \  .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..1000
• Wikipedia, Counting lattice paths

Crossrefs

Column k=2 of A287822.

Cf. A000108.

Programs

Maple

b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> g(n, 2)-g(n, 1):
seq(a(n), n=2..35);

Mathematica

b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j, k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 2] - g[n, 1], {n, 2, 35}] (* Indranil Ghosh, Aug 09 2017 *)

Python

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, k, j): return 1 if j==n else sum(b(n - j, k, i)*sum(binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)) for i in range(1, min(j + k, n - j) + 1))
def g(n, k): return sum(b(n, k, j) for j in range(1, k + 1))
def a(n): return g(n, 2) - g(n, 1)
print([a(n) for n in range(2, 36)]) # Indranil Ghosh, Aug 09 2017