A005202 Total number of fixed points in planted trees with n nodes.

0, 1, 0, 1, 1, 4, 6, 14, 28, 60, 125, 263, 558, 1181, 2513, 5339, 11392, 24290, 51926, 111017, 237757, 509404, 1092713, 2345256, 5038015, 10828720, 23291759, 50126055, 107939753, 232550011, 501270200, 1080996244, 2332221316, 5033764628, 10868950676, 23476998980, 50728408182, 109649040738, 237081174662, 512767906801, 1109354495908

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From R. J. Mathar, Apr 13 2019: (Start)

The associated triangle H_{p,j}, p >= 1, 1 <= j <= p, a(n) = Sum_{j=1..p} j*H_{p,j}, row sums in A001678, starts:

1;

0, 0;

1, 0, 0;

1, 0, 0, 0;

1, 0, 1, 0, 0;

1, 1, 1, 0, 0, 0;

2, 2, 1, 0, 1, 0, 0;

1, 4, 2, 2, 1, 0, 0, 0;

3, 4, 4, 5, 2, 0, 1, 0, 0;

3, 7, 7, 9, 4, 4, 1, 0, 0, 0;

5, 9, 15, 14, 11, 9, 3, 0, 1, 0, 0;

4, 14, 23, 28, 25, 19, 7, 6, 1, 0, 0, 0;

11, 15, 39, 46, 55, 38, 24, 14, 5, 0, 1, 0, 0;

6, 32, 54, 86, 97, 86, 64, 36, 11, 9, 1, 0, 0, 0;

(End)

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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 rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A005200.

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:
hxy := Hxy(x,y,0,0) ;
for pmax from 2 to 20 do
    Hxy(x,y,pmax,hxy) ;
    taylor(%,x=0,pmax+2) ;
    convert(%,polynom) ;
    taylor(%,y=0,pmax+2) ;
    hxy := convert(%,polynom) ;
    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

Mathematica

Hpj[Hofxy_, p_, j_] := SeriesCoefficient[SeriesCoefficient[Hofxy, {x, 0, p}] , {y, 0, j}];
Hxy [x_, y_, pMax_, hxyinit_] := If [pMax == 0, x y, pp = 1; For[p = 1, p <= pMax, p++, t = 1; For[j = 1, j <= p, j++, t = t(1 + x^p y^j + Sum[x^(k*p), {k, 2, pMax + 1}])^Hpj[hxyinit, p, j]]; pp = pp t]; x*y* pp/(1 + x y)];
hxy = Hxy[x, y, 0, 0];
Reap[For[pMax = 2, pMax <= terms - 1, pMax++, Print["pMax = ", pMax]; sx = Series[Hxy[x, y, pMax, hxy], {x, 0, pMax + 2}] // Normal; sy = Series[sx, {y, 0, pMax + 2}]; hxy = sy // Normal; For[p = 0, p <= pMax, p++, Ap = 0; For[j = 1, j <= p, j++, Ap = Ap + j Hpj[hxy, p, j]]; If[pMax == terms - 1, Print[Ap]; Sow[Ap]]]]][[2, 1]] (* Jean-François Alcover, Mar 22 2020, after R. J. Mathar *)

Extensions

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