A355281 Number of pairs of nested Dyck paths from (0,0) to (n,n) such that the upper path only touches the diagonal at its endpoints.

1, 1, 2, 9, 55, 400, 3266, 28999, 274537, 2734885, 28401315, 305352146, 3380956839, 38394091370, 445702108969, 5274935433915, 63507021523471, 776347636736261, 9621502184089320, 120726786082609207, 1531938384684090884, 19639252409244653785, 254143269904958943103, 3317204158078663935592

Offset: 0

Author

Joel B. Lewis, Jun 26 2022

Keywords

• nonn

Comments

Let B be the 2 X n X n box of integer points with opposite corners (0, 0, 0) and (1, n - 1, n - 1). For n >= 1, a(n) is also the number of plane partitions that fit inside B and whose cells lie on or below the plane x + y + z = n - 1. Proof: after rotating by 90 degrees, the upper Dyck path is the outer boundary of the region of the plane partition filled with 2's and the lower Dyck path is the outer boundary of the region of the plane partition filled with 1's or 2's.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..841

Crossrefs

Cf. A000108, A005700.

Column k=2 of A378112.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      b(n-1)*((4*n)^2-4)/(n+2)/(n+3))
    end:
a:= proc(n) option remember;
      b(n)-add(a(n-i)*b(i), i=1..n-1)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 26 2022

Mathematica

nmax = 23;
c = CatalanNumber;
B[x_] = Sum[(c[n] c[n+2] - c[n+1]^2) x^n, {n, 0, nmax}];
CoefficientList[2 - 1/B[x] + O[x]^(nmax+1), x] (* Jean-François Alcover, Jul 06 2022 *)

Formula

G.f.: 2 - 1/B(x) where B(x) is the generating function for A005700.

INVERTi transform of A005700.