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

1,7

Links

Table of n, a(n) for n=1..37.

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

Formula

Reference gives a recurrence.

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

Crossrefs

Cf. A005200-A005202.

Sequence in context: A319130 A329467 A293818 * A310019 A261998 A270413

Adjacent sequences: A037243 A037244 A037245 * A037247 A037248 A037249

Keyword

nonn,easy

Author

N. J. A. Sloane.

Extensions

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

Status

approved