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

0,3

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

Formula

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

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 *)

Crossrefs

Cf. A000108, A005700.

Sequence in context: A267225 A078455 A366177 * A036074 A231172 A009363

Adjacent sequences: A355278 A355279 A355280 * A355282 A355283 A355284

Keyword

nonn

Author

Joel B. Lewis, Jun 26 2022

Status

approved