A037246 - OEIS

%I #10 Apr 13 2019 15:04:01

%S 1,0,0,1,1,1,3,5,10,16,38,66,143,268,564,1100,2282,4546,9382,18977,

%T 39112,79891,164917,339195,702041,1451628,3013442,6257561,13029327,

%U 27152492,56698062,118518363,248137778,520085704,1091520783,2293229235,4823466463

%N Total number of fixed points in free homeomorphically irreducible trees with n nodes.

%H F. Harary and E. M. Palmer, <a href="http://dx.doi.org/10.1017/S0305004100055857">Probability that a point of a tree is fixed</a>, Math. Proc. Camb. Phil. Soc. 85 (1979) 407-415.

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%F Reference gives a recurrence.

%p Hpj := proc(Hofxy,p,j)

%p     coeftayl(Hofxy,x=0,p) ;

%p     coeftayl(%,y=0,j) ;

%p     simplify(%) ;

%p end proc:

%p Hxy := proc(x,y,pmax,hxyinit)

%p     if pmax = 0 then

%p         x*y ;

%p     else

%p         pp := 1;

%p         for p from 1 to pmax do

%p             t :=1 ;

%p             for j from 1 to p do

%p                 t := t*(1+x^p*y^j+add(x^(k*p),k=2..pmax+1))^Hpj(hxyinit,p,j) ;

%p             end do:

%p             pp := pp*t ;

%p         end do:

%p         x*y*%/(1+x*y) ;

%p     end if;

%p end proc:

%p hxyfin := Hxy(x,y,0,0) ;

%p for pmax from 2 to 40 do

%p     Hxy(x,y,pmax,hxyfin) ;

%p     taylor(%,x=0,pmax+2) ;

%p     convert(%,polynom) ;

%p     taylor(%,y=0,pmax+2) ;

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

%p     hxy := (1+x*y)*hxyfin+subs({x=x^2,y=1},hxyfin)*(1-x*y)-hxyfin^2*(1+x*y)/2+subs({x=x^2,y=y^2},hxyfin)*(x*y-1)/2 ;

%p     for p from 0 to pmax do

%p         ap := 0 ;

%p         for j from 1 to p do

%p             ap := ap+j*Hpj(hxy,p,j) ;

%p         end do:

%p         printf("%d,",ap) ;

%p     end do:

%p     print() ;

%p end do: # _R. J. Mathar_, Apr 13 2019

%Y Cf. A005200-A005202.

%K nonn,easy

%O 1,7

%A _N. J. A. Sloane_.

%E More terms from _R. J. Mathar_, Apr 13 2019