A391955 Number of pairs of Dyck paths of length 2*n touching the axis at the same points.

1, 1, 2, 7, 38, 274, 2350, 22531, 233292, 2555658, 29232554, 346013450, 4211121946, 52446977292, 666024794758, 8599676755883, 112647192598844, 1494224720878614, 20041069061550880, 271454315346852530, 3709291397981162290, 51088066055510683620

Offset

0,3

Comments

a(n) is the number of congruences of any biCambrian quotient of A_{n-1} (for example the lattice of Baxter permutations of length n).

Links

Table of n, a(n) for n=0..21.

Emily Barnard and Nathan Reading, Coxeter-biCatalan combinatorics, arXiv:1605.03524 [math.CO], 2016-2017.

Formula

G.f.: 1/(1-x*S) where S = 1 + x + 4*x^2 + 25*x^3... is the g.f. for A001246.

G.f.: 4/(5 - hypergeom([-1/2, -1/2], [1], 16*x)). - Stefano Spezia, Dec 24 2025

Example

The a(3) = 7 pairs of Dyck paths of length 3 touching the axis at the same points are:
     /\        /\
    /  \      /  \      /\/\      /\/\      /\          /\
   /    \    /    \    /    \    /    \    /  \/\    /\/  \    /\/\/\
   \    /    \    /    \    /    \    /    \  /\/    \/\  /    \/\/\/
    \  /      \/\/      \  /      \/\/      \/          \/
     \/                  \/

Mathematica

CoefficientList[Series[4/(5-Hypergeometric2F1[- 1/2, -1/2, 1, 16x]), {x, 0, 21}], x] (* Stefano Spezia, Dec 24 2025 *)

Crossrefs

Cf. A000108, A001181, A001246.

Sequence in context: A381572 A032109 A368232 * A337026 A088792 A114160

Adjacent sequences: A391952 A391953 A391954 * A391956 A391957 A391958

Keyword

nonn

Author

Ludovic Schwob, Dec 23 2025

Status

approved