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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, 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
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
Sequence in context: A267225 A078455 A366177 * A036074 A231172 A009363
KEYWORD
nonn
AUTHOR
Joel B. Lewis, Jun 26 2022
STATUS
approved