A291885 Number of symmetrically unique Dyck paths of semilength 2n and height n.

1, 1, 5, 31, 252, 2117, 18546, 164229, 1469596, 13229876, 119712521, 1087573357, 9914033252, 90633332870, 830621140260, 7628813061585, 70200092854044, 647070588612140, 5973385906039684, 55217660246861884, 511054426374819184, 4735208302827742549

Offset: 0

Author

Alois P. Heinz, Sep 05 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1026

Crossrefs

Cf. A291883.

Programs

Maple

b:= proc(x, y, k) option remember; `if`(x=0, 1, `if`(y+1<=min(k,
        x-1), b(x-1, y+1, k), 0)+`if`(y>0, b(x-1, y-1, k), 0))
    end:
g:= proc(x, y, k) option remember; `if`(x=0, 1, `if`(y>0,
      g(x-2, y-1, k), 0)+ `if`(y+1<=k, g(x-2, y+1, k), 0))
    end:
a:= n-> `if`(n=0, 1, (b(4*n, 0, n)  +g(4*n, 0, n)
                     -b(4*n, 0, n-1)-g(4*n, 0, n-1))/2):
seq(a(n), n=0..30);

Mathematica

b[x_, y_, k_] := b[x, y, k] = If[x == 0, 1, If[y + 1 <= Min[k, x - 1], b[x - 1, y + 1, k], 0] + If[y > 0, b[x - 1, y - 1, k], 0]];
g[x_, y_, k_] := g[x, y, k] = If[x == 0, 1, If[y > 0, g[x - 2, y - 1, k], 0] + If[y + 1 <= k, g[x - 2, y + 1, k], 0]];
a[n_] := If[n == 0, 1, (b[4n, 0, n] + g[4n, 0, n] - b[4n, 0, n - 1] - g[4n, 0, n - 1])/2];
Array[a, 30, 0] (* Jean-François Alcover, May 31 2019, after Alois P. Heinz *)

Python

from sympy.core.cache import cacheit
@cacheit
def b(x, y, k): return 1 if x==0 else (b(x - 1, y + 1, k) if y + 1<=min(k, x - 1) else 0) + (b(x - 1, y - 1, k) if y>0 else 0)
@cacheit
def g(x, y, k): return 1 if x==0 else (g(x - 2, y - 1, k) if y>0 else 0) + (g(x - 2, y + 1, k) if y + 1<=k else 0)
def a(n): return 1 if n==0 else (b(4*n, 0, n) + g(4*n, 0, n) - b(4*n, 0, n - 1) - g(4*n, 0, n - 1))//2
print([a(n) for n in range(31)]) # Indranil Ghosh, Sep 06 2017

Formula

a(n) = A291883(2n,n).