A389288 Number of nested pairs of Dyck paths of lengths n and n+1.

1, 2, 8, 44, 294, 2244, 18876, 171028, 1643356, 16560856, 173603456, 1881260528, 20972215580, 239589707400, 2796187194900, 33252827904900, 402099798424500, 4935245946978600, 61390057645778400, 772931874857828400, 9839101208224883400

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = 6 * (6*n+4) * (2*n)! * (2*n+2)! / (n! * (n+1)! * (n+2)! * (n+4)!).

a(n) ~ 2187*16^(4+n)/((1 + 12*n)*(25 + 312*n + 288*n^2)*(649 + 888*n + 288*n^2)*Pi). - Stefano Spezia, Sep 29 2025

Example

The a(2) = 8 pairs of Dyck paths are:
             /\          /\      /\/\
  /\/\/\    /  \/\    /\/  \    /    \
  /\/\      /\/\      /\/\      /\/\
.
    /\                            /\
   /  \      /\        /\/\      /  \
  /    \    //\\/\    //\  \    //\  \
  /\/\      /  \      /  \      /  \
.

Mathematica

Table[6 * (6*n+4) * Factorial(2*n) * Factorial(2*n+2) / (Factorial(n) * Factorial(n+1) * Factorial(n+2) * Factorial(n+4)), {n, 0, 30}] (* Vasily V. Zolotukhin, Sep 29 2025 *)

Prog

(Magma) [6 * (6*n+4) * Factorial(2*n) * Factorial(2*n+2) / (Factorial(n) * Factorial(n+1) * Factorial(n+2) * Factorial(n+4)): n in [0..30]]; // Vasily V. Zolotukhin, Sep 29 2025
(SageMath) [6 * (6*n+4) * factorial(2*n) * factorial(2*n+2) / (factorial(n) * factorial(n+1) * factorial(n+2) * factorial(n+4)) for n in (0..30)] # Vasily V. Zolotukhin, Sep 29 2025

Crossrefs

Cf. A000108, A005700.

Sequence in context: A240165 A357832 A318977 * A111537 A051045 A112912

Adjacent sequences: A389285 A389286 A389287 * A389289 A389290 A389291

Keyword

nonn,easy

Author

Ludovic Schwob, Sep 28 2025

Status

approved