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URL: https://oeis.org/A291885

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A291885
Number of symmetrically unique Dyck paths of semilength 2n and height n.
2
1, 1, 5, 31, 252, 2117, 18546, 164229, 1469596, 13229876, 119712521, 1087573357, 9914033252, 90633332870, 830621140260, 7628813061585, 70200092854044, 647070588612140, 5973385906039684, 55217660246861884, 511054426374819184, 4735208302827742549
OFFSET
0,3
LINKS
FORMULA
a(n) = A291883(2n,n).
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 *)
PROG
(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
CROSSREFS
Cf. A291883.
Sequence in context: A375533 A361408 A056541 * A126121 A167137 A279434
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 05 2017
STATUS
approved