A322379 Triangle T(s,d) read by rows: the number of 2-connected labeled cubic graphs with s simple edges and d double edges.

0, 0, 0, 0, 0, 6, 0, 0, 0, 120, 0, 0, 0, 0, 5040, 0, 0, 180, 0, 0, 362880, 1, 0, 0, 23520, 0, 0, 39916800, 0, 180, 0, 0, 3628800, 0, 0, 6227020800, 0, 0, 45360, 0, 0, 718502400, 0, 0, 1307674368000, 70, 0, 0, 13003200, 0, 0, 181621440000, 0, 0, 355687428096000, 0, 45360, 0, 0, 4340952000, 0, 0, 57537672192000, 0, 0

Offset

0,6

Links

Table of n, a(n) for n=0..64.

G.-B. Chae, E. M. Palmer, R. W. Robinson, Counting labeled cubic graphs, Disc. Math. 307 (2007) 2979-2992, g(s,d).

Formula

T(3s,0) = A007099(s).

Example

The triangle starts
  0;
  0, 0;
  0, 0,  6;
  0, 0,  0,   120;
  0, 0,  0,     0, 5040;
  0, 0,180,     0,    0, 362880;
  1, 0,  0, 23520,    0,      0, 39916800;

Maple

# expand g(s, d) of eq (21) of Chae et al.
g2x := 6*x^5/4! ;
for itr from 1 to 16 do
    g2xx := expand(diff(g2x, x)) ;
    g2x := (x^5-x^8)*g2x*g2xx+(x^4-2*x^7+x^10+x^5*y-x^8*y)/2*g2xx
        +(2*x^4+x^7)*g2x^2
        +(8*x^3-6*x^6-x^9+x^12+2*x*y-2*x^4*y+8*x^7*y-2*x^10*y)/2*g2x
        + x^5/4 -3*x^8/4 +3*x^11/4-x^14/4 +3*x^6*y/2-9*x^9*y/4+3*x^12*y/4+x*y^2/2
        -x^4*y^2+7*x^7*y^2/4-x^10*y^2/2 ;
    g2x := expand(%) ;
    g2x := taylor(g2x, x=0, itr+5) ;
    g2x := convert(g2x, polynom) ;
    g2 := expand(int(g2x, x)) ;
    for s from 0 to itr+1 do
        g := coeftayl(g2, x=0, s) ;
        for d from 0 to s do
            twon := (2*s+4*d)/3 ;
            coeftayl(g, y=0, d) ;
            printf("%a, ", %*twon!) ;
        end do:
        printf("\n") ;
    end do:
end do:

Crossrefs

Cf. A009445 (diagonal), A007099 (left column).

Sequence in context: A096527 A028599 A270850 * A225339 A365989 A365990

Adjacent sequences: A322376 A322377 A322378 * A322380 A322381 A322382

Keyword

nonn,tabl

Author

R. J. Mathar, Dec 05 2018

Status

approved