cp's OEIS Frontend

A037246 Total number of fixed points in free homeomorphically irreducible trees with n nodes.

%I A037246 #12 Jul 08 2025 22:05:35
%S A037246 1,0,0,1,1,1,3,5,10,16,38,66,143,268,564,1100,2282,4546,9382,18977,
%T A037246 39112,79891,164917,339195,702041,1451628,3013442,6257561,13029327,
%U A037246 27152492,56698062,118518363,248137778,520085704,1091520783,2293229235,4823466463
%N A037246 Total number of fixed points in free homeomorphically irreducible trees with n nodes.
%H A037246 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 A037246 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A037246 Reference gives a recurrence.
%p A037246 Hpj := proc(Hofxy,p,j)
%p A037246     coeftayl(Hofxy,x=0,p) ;
%p A037246     coeftayl(%,y=0,j) ;
%p A037246     simplify(%) ;
%p A037246 end proc:
%p A037246 Hxy := proc(x,y,pmax,hxyinit)
%p A037246     if pmax = 0 then
%p A037246         x*y ;
%p A037246     else
%p A037246         pp := 1;
%p A037246         for p from 1 to pmax do
%p A037246             t :=1 ;
%p A037246             for j from 1 to p do
%p A037246                 t := t*(1+x^p*y^j+add(x^(k*p),k=2..pmax+1))^Hpj(hxyinit,p,j) ;
%p A037246             end do:
%p A037246             pp := pp*t ;
%p A037246         end do:
%p A037246         x*y*%/(1+x*y) ;
%p A037246     end if;
%p A037246 end proc:
%p A037246 hxyfin := Hxy(x,y,0,0) ;
%p A037246 for pmax from 2 to 40 do
%p A037246     Hxy(x,y,pmax,hxyfin) ;
%p A037246     taylor(%,x=0,pmax+2) ;
%p A037246     convert(%,polynom) ;
%p A037246     taylor(%,y=0,pmax+2) ;
%p A037246     hxyfin := convert(%,polynom) ;
%p A037246     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 A037246     for p from 0 to pmax do
%p A037246         ap := 0 ;
%p A037246         for j from 1 to p do
%p A037246             ap := ap+j*Hpj(hxy,p,j) ;
%p A037246         end do:
%p A037246         printf("%d,",ap) ;
%p A037246     end do:
%p A037246     print() ;
%p A037246 end do: # _R. J. Mathar_, Apr 13 2019
%Y A037246 Cf. A005200-A005202.
%K A037246 nonn,easy
%O A037246 1,7
%A A037246 _N. J. A. Sloane_
%E A037246 More terms from _R. J. Mathar_, Apr 13 2019