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%I #26 Feb 15 2024 19:06:43
%S 1,6,71,1266,30206,902796,32420011,1359292626,65164480466,
%T 3515569641156,210779736073446,13903319821066836,1000559812125494076,
%U 78012524487061315416,6550837823204594551731,589404446176366002280146,56568586570039148217467786,5768723174387469795772704276,622900652040379217092492454866
%N Number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))].
%C The definition of a one-level grid poset can be found in the Pan links. The number of linear extensions of the one-level grid poset G[(0^n), (0^(n-1)), (0^(n-1))] is given by Catalan number A000108(n).
%H Michael Wallner, <a href="/A274644/b274644.txt">Table of n, a(n) for n = 1..100</a>
%H Cyril Banderier and Michael Wallner, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2021/47.html">Young Tableaux with Periodic Walls: Counting with the Density Method</a>, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.
%H Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/problems/p1.html">Problem 1</a>, Project P.
%H Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/problems/solutions/OneLevelGridPoset.pdf">Algorithmic Solution to Problem 1 (and linear extensions of general one-level grid-like posets)</a>, Project P.
%F a(n) = b(n,3) in b(n,k) = Sum_{1 <= i <=k} i*b(n-1,i+2) for n>0 and k>=3 with initial conditions b(1,k)=1 for all k. - _Michael Wallner_, Feb 13 2024
%F a(n) = (3*n)!*int(int(f_{n}(x,y),x=0..y),y=0..1) where f_{n+1} = (y-x)*int(int(f_{n}(v,w)),w=v..y),v=0..x)) for n>=1 and f_{1}(x,y) = y-x (Derived using the density method; see [Banderier, Wallner 2021]). - _Michael Wallner_, Feb 13 2024
%p M := 20;
%p for k from 3 to 3+2*M do
%p bb[1,k] := 1;
%p end:
%p for n from 2 to M do
%p for k from 3 to 3+2*M-2*(n-1) do
%p bb[n,k] := sum(i*bb[n-1,i+2],i=1..k);
%p end;
%p end:
%p seq(bb[n,3],n=1..10);
%p N := 100:
%p f[1] := y-x;
%p for n from 1 to N-1 do
%p f[n+1] := (y-x)*int(int(subs(x=v,y=w,f[n]),w=v..y),v=0..x);
%p end:
%p for n from 1 to N do
%p aa[n] := factorial(3*n)*int(int(f[n],x=0..y),y=0..1);
%p end:
%p seq(aa[n],n=1..10);
%p # _Michael Wallner_, Feb 13 2024
%Y Cf. A000108, A274763.
%K nonn
%O 1,2
%A _Ran Pan_, Jun 30 2016
%E All terms starting with a(13) corrected by _Michael Wallner_, Feb 13 2024
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