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A296419 Triangle T(i,j) read by rows: Number of plane bipolar orientations with i+1 vertices and j+1 faces.

%I A296419 #15 Jun 25 2023 08:10:41
%S A296419 1,1,4,1,10,50,1,20,175,980,1,35,490,4116,24696,1,56,1176,14112,
%T A296419 116424,731808,1,84,2520,41580,457380,3737448,24293412,1,120,4950,
%U A296419 108900,1557270,16195608,131589315,877262100,1,165,9075,259545,4723719,61408347,614083470,4971151900,33803832920
%N A296419 Triangle T(i,j) read by rows: Number of plane bipolar orientations with i+1 vertices and j+1 faces.
%H A296419 E. Fusy, D. Poulalhon, and G. Schaeffer, <a href="https://doi.org/10.1016/j.endm.2007.07.049">Bijective counting of plane bipolar orientations</a>, El. Notes Discr. Math. 29 (2007) 283-287.
%F A296419 T(i,j) = T(j,i) = 2*(i+j-2)!*(i+j-1)!*(i+j)!/((i-1)!*i!*(i+1)!*(j-1)!*j!*(j+1)!).
%e A296419 The triangle starts in row 1 as
%e A296419   1;
%e A296419   1,   4;
%e A296419   1,  10,   50;
%e A296419   1,  20,  175,    980;
%e A296419   1,  35,  490,   4116,   24696;
%e A296419   1,  56, 1176,  14112,  116424,   731808;
%e A296419   1,  84, 2520,  41580,  457380,  3737448,  24293412;
%e A296419   1, 120, 4950, 108900, 1557270, 16195608, 131589315, 877262100;
%p A296419 A296419 := proc(i,j)
%p A296419     2*(i+j-2)!*(i+j-1)!*(i+j)!/(i-1)!/i!/(i+1)!/(j-1)!/j!/(j+1)! ;
%p A296419 end proc:
%p A296419 seq(seq(A296419(i,j),j=1..i),i=1..10) ;
%Y A296419 Cf. rows/columns: A006542, A047819, A107915, A140901, A140903, A140907.
%K A296419 nonn,tabl,easy
%O A296419 1,3
%A A296419 _R. J. Mathar_, Feb 25 2018