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

1, 1, 4, 1, 10, 50, 1, 20, 175, 980, 1, 35, 490, 4116, 24696, 1, 56, 1176, 14112, 116424, 731808, 1, 84, 2520, 41580, 457380, 3737448, 24293412, 1, 120, 4950, 108900, 1557270, 16195608, 131589315, 877262100, 1, 165, 9075, 259545, 4723719, 61408347, 614083470, 4971151900, 33803832920

Offset

1,3

Links

Table of n, a(n) for n=1..45.

E. Fusy, D. Poulalhon, and G. Schaeffer, Bijective counting of plane bipolar orientations, El. Notes Discr. Math. 29 (2007) 283-287.

Formula

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)!).

Example

The triangle starts in row 1 as
  1;
  1,   4;
  1,  10,   50;
  1,  20,  175,    980;
  1,  35,  490,   4116,   24696;
  1,  56, 1176,  14112,  116424,   731808;
  1,  84, 2520,  41580,  457380,  3737448,  24293412;
  1, 120, 4950, 108900, 1557270, 16195608, 131589315, 877262100;

Maple

A296419 := proc(i, j)
    2*(i+j-2)!*(i+j-1)!*(i+j)!/(i-1)!/i!/(i+1)!/(j-1)!/j!/(j+1)! ;
end proc:
seq(seq(A296419(i, j), j=1..i), i=1..10) ;

Crossrefs

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

Sequence in context: A307529 A019213 A019128 * A301390 A283433 A121463

Adjacent sequences: A296416 A296417 A296418 * A296420 A296421 A296422

Keyword

nonn,tabl,easy

Author

R. J. Mathar, Feb 25 2018

Status

approved