A182530 Number of Dyck 2n-paths with all ascents of even length and all descents of odd length.

1, 0, 1, 3, 9, 33, 125, 484, 1933, 7883, 32656, 137127, 582353, 2496711, 10791823, 46978411, 205775845, 906291623, 4011033078, 17829430207, 79565177628, 356329040864, 1600966438342, 7214304154481, 32597284112813, 147655425387293, 670371400288210

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Vaclav Kotesovec, Recurrence (of order 8)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 4.80922177566836796327406959229376... is the root of the equation 20 - 223*d - 36*d^2 - 60*d^3 + 16*d^4 = 0, c = 0.1691900578944028281094160273139... . - Vaclav Kotesovec, Aug 22 2014

Example

For n=3 the three Dyck 6-paths are UUUUDUUDDDDD, UUUUDDDUUDDD and UUDUUUUDDDDD.

Maple

b:= proc(x, y, d) option remember;
      `if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1,
       b(x, y-2, true) +b(x-`if`(d, 1, 2), y, false)))
    end:
a:= n-> b(2*n, 2*n, true):
seq(a(n), n=0..30);  # Alois P. Heinz, May 04 2012

Mathematica

NN=26; Array[a, {NN*2+1, NN*2+1}]; seq={};
fodd[t_, k_]:=a[t, k]=If[k==1, 1, If[k==Floor[t/2], a[t, k-1]+a[t-1, k-1]+a[t-3, k-1], a[t-2, k]+a[t, k-1]+a[t-1, k-1]-a[t-2, k-1]]];
feven[t_, k_]:=a[t, k]=If[k==1, 0, If[k==Floor[t/2], a[t, k-1]+a[t-1, k-1], a[t-2, k]+a[t, k-1]+a[t-1, k-1]-a[t-2, k-1]]];
For[t=2, t<=NN*2, t++, For[k=1, k<=Floor[t/2], k++, If[OddQ[t], fodd[t, k], feven[t, k]; If[k==Floor[t/2], seq=Append[seq, a[t, k]]]]]]; seq
(* Second program: *)
b[x_, y_, d_] := b[x, y, d] =
     If[x < 0 || y < x, 0, If[x == 0 && y == 0, 1,
     b[x, y - 2, True] + b[x - If[d, 1, 2], y, False]]];
a[n_] := b[2*n, 2*n, True];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz *)

Crossrefs

Sequence in context: A039648 A307454 A219261 * A372577 A049182 A372530

Adjacent sequences: A182527 A182528 A182529 * A182531 A182532 A182533

Keyword

nonn

Author

David Scambler, May 03 2012

Status

approved