# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a005202 Showing 1-1 of 1 %I A005202 M3282 #21 Mar 22 2020 10:01:28 %S A005202 0,1,0,1,1,4,6,14,28,60,125,263,558,1181,2513,5339,11392,24290,51926, %T A005202 111017,237757,509404,1092713,2345256,5038015,10828720,23291759, %U A005202 50126055,107939753,232550011,501270200,1080996244,2332221316,5033764628,10868950676,23476998980,50728408182,109649040738,237081174662,512767906801,1109354495908 %N A005202 Total number of fixed points in planted trees with n nodes. %C A005202 From _R. J. Mathar_, Apr 13 2019: (Start) %C A005202 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: %C A005202 1; %C A005202 0, 0; %C A005202 1, 0, 0; %C A005202 1, 0, 0, 0; %C A005202 1, 0, 1, 0, 0; %C A005202 1, 1, 1, 0, 0, 0; %C A005202 2, 2, 1, 0, 1, 0, 0; %C A005202 1, 4, 2, 2, 1, 0, 0, 0; %C A005202 3, 4, 4, 5, 2, 0, 1, 0, 0; %C A005202 3, 7, 7, 9, 4, 4, 1, 0, 0, 0; %C A005202 5, 9, 15, 14, 11, 9, 3, 0, 1, 0, 0; %C A005202 4, 14, 23, 28, 25, 19, 7, 6, 1, 0, 0, 0; %C A005202 11, 15, 39, 46, 55, 38, 24, 14, 5, 0, 1, 0, 0; %C A005202 6, 32, 54, 86, 97, 86, 64, 36, 11, 9, 1, 0, 0, 0; %C A005202 (End) %D A005202 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A005202 F. Harary and E. M. Palmer, Probability that a point of a tree is fixed, Math. Proc. Camb. Phil. Soc. 85 (1979) 407-415. %H A005202 Index entries for sequences related to rooted trees %H A005202 Index entries for sequences related to trees %p A005202 Hpj := proc(Hofxy,p,j) %p A005202 coeftayl(Hofxy,x=0,p) ; %p A005202 coeftayl(%,y=0,j) ; %p A005202 simplify(%) ; %p A005202 end proc: %p A005202 Hxy := proc(x,y,pmax,hxyinit) %p A005202 if pmax = 0 then %p A005202 x*y ; %p A005202 else %p A005202 pp := 1; %p A005202 for p from 1 to pmax do %p A005202 t :=1 ; %p A005202 for j from 1 to p do %p A005202 t := t*(1+x^p*y^j+add(x^(k*p),k=2..pmax+1))^Hpj(hxyinit,p,j) ; %p A005202 end do: %p A005202 pp := pp*t ; %p A005202 end do: %p A005202 x*y*%/(1+x*y) ; %p A005202 end if; %p A005202 end proc: %p A005202 hxy := Hxy(x,y,0,0) ; %p A005202 for pmax from 2 to 20 do %p A005202 Hxy(x,y,pmax,hxy) ; %p A005202 taylor(%,x=0,pmax+2) ; %p A005202 convert(%,polynom) ; %p A005202 taylor(%,y=0,pmax+2) ; %p A005202 hxy := convert(%,polynom) ; %p A005202 for p from 0 to pmax do %p A005202 Ap := 0 ; %p A005202 for j from 1 to p do %p A005202 Ap := Ap+j*Hpj(hxy,p,j) ; %p A005202 end do: %p A005202 printf("%d,",Ap) ; %p A005202 end do: %p A005202 print() ; %p A005202 end do: # _R. J. Mathar_, Apr 13 2019 %t A005202 Hpj[Hofxy_, p_, j_] := SeriesCoefficient[SeriesCoefficient[Hofxy, {x, 0, p}] , {y, 0, j}]; %t A005202 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)]; %t A005202 hxy = Hxy[x, y, 0, 0]; %t A005202 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_ *) %Y A005202 Cf. A005200. %K A005202 nonn,easy %O A005202 1,6 %A A005202 _N. J. A. Sloane_ %E A005202 More terms from _R. J. Mathar_, Apr 13 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE