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Triangle T(i,j) read by rows: Number of plane bipolar orientations with i+1 vertices and j+1 faces.
3

%I #15 Jun 25 2023 08:10:41

%S 1,1,4,1,10,50,1,20,175,980,1,35,490,4116,24696,1,56,1176,14112,

%T 116424,731808,1,84,2520,41580,457380,3737448,24293412,1,120,4950,

%U 108900,1557270,16195608,131589315,877262100,1,165,9075,259545,4723719,61408347,614083470,4971151900,33803832920

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

%H 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 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 The triangle starts in row 1 as

%e 1;

%e 1, 4;

%e 1, 10, 50;

%e 1, 20, 175, 980;

%e 1, 35, 490, 4116, 24696;

%e 1, 56, 1176, 14112, 116424, 731808;

%e 1, 84, 2520, 41580, 457380, 3737448, 24293412;

%e 1, 120, 4950, 108900, 1557270, 16195608, 131589315, 877262100;

%p A296419 := proc(i,j)

%p 2*(i+j-2)!*(i+j-1)!*(i+j)!/(i-1)!/i!/(i+1)!/(j-1)!/j!/(j+1)! ;

%p end proc:

%p seq(seq(A296419(i,j),j=1..i),i=1..10) ;

%Y Cf. rows/columns: A006542, A047819, A107915, A140901, A140903, A140907.

%K nonn,tabl,easy

%O 1,3

%A _R. J. Mathar_, Feb 25 2018