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%I #31 Aug 18 2023 15:59:46
%S 1,1,1,2,5,12,32,93,279,872,2830,9433,32223,112527,400370,1448520,
%T 5320023,19802827,74612164,284238390,1093757436,4247742956,
%U 16636921148,65671960544,261111950308,1045172796381,4209807155949,17055625810984,69476952146529,284467866640048
%N Number of plane bipolar posets (i.e., plane bipolar orientations with no transitive edge) with n edges.
%C a(n) is also the number of walks of length n-1 in the quadrant, starting and ending at the origin, with step-set {0,E,S,NW,SE} (where 0 is the stay-step).
%H Éric Fusy, Erkan Narmanli, and Gilles Schaeffer, <a href="https://arxiv.org/abs/2105.06955">On the enumeration of plane bipolar posets and transversal structures</a>, arXiv:2105.06955 [math.CO], 2021-2022.
%p A:=proc(n,i,j) option remember:
%p if n=0 and i=0 and j=0 then return 1:
%p elif n<=0 or j<0 or i<0 then return 0:
%p else
%p return A(n-1,i,j)+A(n-1,i-1,j)+A(n-1,i,j+1)+A(n-1,i+1,j-1)+A(n-1,i-1,j+1):
%p fi:
%p end proc:
%p seq(A(n-1,0,0),n=1..20);
%Y Cf. A001181, A296419, A117106, A151335.
%K nonn
%O 1,4
%A _Éric Fusy_, Jun 16 2023
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