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
Examples
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;
Links
- G.-B. Chae, E. M. Palmer, R. W. Robinson, Counting labeled cubic graphs, Disc. Math. 307 (2007) 2979-2992, g(s,d).
Programs
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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:
Formula
T(3s,0) = A007099(s).