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A005202 Total number of fixed points in planted trees with n nodes.
(Formerly M3282)
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

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

Table of n, a(n) for n=1..41.

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

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-Fran├žois Alcover, Mar 22 2020, after R. J. Mathar *)

CROSSREFS

Cf. A005200.

Sequence in context: A009849 A303041 A103419 * A106526 A319766 A108516

Adjacent sequences:  A005199 A005200 A005201 * A005203 A005204 A005205

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified August 7 17:44 EDT 2020. Contains 336278 sequences. (Running on oeis4.)