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

1, 0, 0, 1, 1, 1, 3, 5, 10, 16, 38, 66, 143, 268, 564, 1100, 2282, 4546, 9382, 18977, 39112, 79891, 164917, 339195, 702041, 1451628, 3013442, 6257561, 13029327, 27152492, 56698062, 118518363, 248137778, 520085704, 1091520783, 2293229235, 4823466463

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• F. Harary and E. M. Palmer, Probability that a point of a tree is fixed, Math. Proc. Camb. Phil. Soc. 85 (1979) 407-415.
• Index entries for sequences related to trees

Crossrefs

Cf. A005200-A005202.

Programs

Maple

Hpj := proc(Hofxy,p,j)
    coeftayl(Hofxy,x=0,p) ;
    coeftayl(%,y=0,j) ;
    simplify(%) ;
end proc:
Hxy := proc(x,y,pmax,hxyinit)
    if pmax = 0 then
        x*y ;
    else
        pp := 1;
        for p from 1 to pmax do
            t :=1 ;
            for j from 1 to p do
                t := t*(1+x^p*y^j+add(x^(k*p),k=2..pmax+1))^Hpj(hxyinit,p,j) ;
            end do:
            pp := pp*t ;
        end do:
        x*y*%/(1+x*y) ;
    end if;
end proc:
hxyfin := Hxy(x,y,0,0) ;
for pmax from 2 to 40 do
    Hxy(x,y,pmax,hxyfin) ;
    taylor(%,x=0,pmax+2) ;
    convert(%,polynom) ;
    taylor(%,y=0,pmax+2) ;
    hxyfin := convert(%,polynom) ;
    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 ;
    for p from 0 to pmax do
        ap := 0 ;
        for j from 1 to p do
            ap := ap+j*Hpj(hxy,p,j) ;
        end do:
        printf("%d,",ap) ;
    end do:
    print() ;
end do: # R. J. Mathar, Apr 13 2019

Formula

Reference gives a recurrence.

Extensions

More terms from R. J. Mathar, Apr 13 2019