cp's OEIS Frontend

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.

%I A355281 #28 Nov 16 2024 19:15:40
%S A355281 1,1,2,9,55,400,3266,28999,274537,2734885,28401315,305352146,
%T A355281 3380956839,38394091370,445702108969,5274935433915,63507021523471,
%U A355281 776347636736261,9621502184089320,120726786082609207,1531938384684090884,19639252409244653785,254143269904958943103,3317204158078663935592
%N 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.
%C A355281 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.
%H A355281 Alois P. Heinz, <a href="/A355281/b355281.txt">Table of n, a(n) for n = 0..841</a>
%F A355281 G.f.: 2 - 1/B(x) where B(x) is the generating function for A005700.
%F A355281 INVERTi transform of A005700.
%p A355281 b:= proc(n) option remember; `if`(n=0, 1,
%p A355281       b(n-1)*((4*n)^2-4)/(n+2)/(n+3))
%p A355281     end:
%p A355281 a:= proc(n) option remember;
%p A355281       b(n)-add(a(n-i)*b(i), i=1..n-1)
%p A355281     end:
%p A355281 seq(a(n), n=0..23);  # _Alois P. Heinz_, Jun 26 2022
%t A355281 nmax = 23;
%t A355281 c = CatalanNumber;
%t A355281 B[x_] = Sum[(c[n] c[n+2] - c[n+1]^2) x^n, {n, 0, nmax}];
%t A355281 CoefficientList[2 - 1/B[x] + O[x]^(nmax+1), x] (* _Jean-François Alcover_, Jul 06 2022 *)
%Y A355281 Cf. A000108, A005700.
%Y A355281 Column k=2 of A378112.
%K A355281 nonn
%O A355281 0,3
%A A355281 _Joel B. Lewis_, Jun 26 2022