A322379 - OEIS

%I #14 Feb 12 2021 12:11:54

%S 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,

%T 0,39916800,0,180,0,0,3628800,0,0,6227020800,0,0,45360,0,0,718502400,

%U 0,0,1307674368000,70,0,0,13003200,0,0,181621440000,0,0,355687428096000,0,45360,0,0,4340952000,0,0,57537672192000,0,0

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

%H G.-B. Chae, E. M. Palmer, R. W. Robinson, <a href="https://doi.org/10.1016/j.disc.2007.03.011">Counting labeled cubic graphs</a>, Disc. Math. 307 (2007) 2979-2992, g(s,d).

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

%e The triangle starts

%e   0;

%e   0, 0;

%e   0, 0,  6;

%e   0, 0,  0,   120;

%e   0, 0,  0,     0, 5040;

%e   0, 0,180,     0,    0, 362880;

%e   1, 0,  0, 23520,    0,      0, 39916800;

%p # expand g(s,d) of eq (21) of Chae et al.

%p g2x := 6*x^5/4! ;

%p for itr from 1 to 16 do

%p     g2xx := expand(diff(g2x,x)) ;

%p     g2x := (x^5-x^8)*g2x*g2xx+(x^4-2*x^7+x^10+x^5*y-x^8*y)/2*g2xx

%p         +(2*x^4+x^7)*g2x^2

%p         +(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

%p         + 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

%p         -x^4*y^2+7*x^7*y^2/4-x^10*y^2/2 ;

%p     g2x := expand(%) ;

%p     g2x := taylor(g2x,x=0,itr+5) ;

%p     g2x := convert(g2x,polynom) ;

%p     g2 := expand(int(g2x,x)) ;

%p     for s from 0 to itr+1 do

%p         g := coeftayl(g2,x=0,s) ;

%p         for d from 0 to s do

%p             twon := (2*s+4*d)/3 ;

%p             coeftayl(g,y=0,d) ;

%p             printf("%a,",%*twon!) ;

%p         end do:

%p         printf("\n") ;

%p     end do:

%p end do:

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

%K nonn,tabl

%O 0,6

%A _R. J. Mathar_, Dec 05 2018