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

%I A291885 #21 Apr 26 2021 08:01:20
%S A291885 1,1,5,31,252,2117,18546,164229,1469596,13229876,119712521,1087573357,
%T A291885 9914033252,90633332870,830621140260,7628813061585,70200092854044,
%U A291885 647070588612140,5973385906039684,55217660246861884,511054426374819184,4735208302827742549
%N A291885 Number of symmetrically unique Dyck paths of semilength 2n and height n.
%H A291885 Alois P. Heinz, <a href="/A291885/b291885.txt">Table of n, a(n) for n = 0..1026</a>
%F A291885 a(n) = A291883(2n,n).
%p A291885 b:= proc(x, y, k) option remember; `if`(x=0, 1, `if`(y+1<=min(k,
%p A291885         x-1), b(x-1, y+1, k), 0)+`if`(y>0, b(x-1, y-1, k), 0))
%p A291885     end:
%p A291885 g:= proc(x, y, k) option remember; `if`(x=0, 1, `if`(y>0,
%p A291885       g(x-2, y-1, k), 0)+ `if`(y+1<=k, g(x-2, y+1, k), 0))
%p A291885     end:
%p A291885 a:= n-> `if`(n=0, 1, (b(4*n, 0, n)  +g(4*n, 0, n)
%p A291885                      -b(4*n, 0, n-1)-g(4*n, 0, n-1))/2):
%p A291885 seq(a(n), n=0..30);
%t A291885 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]];
%t A291885 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]];
%t A291885 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];
%t A291885 Array[a, 30, 0] (* _Jean-François Alcover_, May 31 2019, after _Alois P. Heinz_ *)
%o A291885 (Python)
%o A291885 from sympy.core.cache import cacheit
%o A291885 @cacheit
%o A291885 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)
%o A291885 @cacheit
%o A291885 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)
%o A291885 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
%o A291885 print([a(n) for n in range(31)]) # _Indranil Ghosh_, Sep 06 2017
%Y A291885 Cf. A291883.
%K A291885 nonn
%O A291885 0,3
%A A291885 _Alois P. Heinz_, Sep 05 2017