A005200 Total number of fixed points in rooted trees with n nodes.
1, 2, 4, 11, 28, 78, 213, 598, 1670, 4723, 13356, 37986, 108193, 309169, 884923, 2538369, 7292170, 20982220, 60451567, 174385063, 503600439, 1455827279, 4212464112, 12199373350, 35357580112, 102552754000, 297651592188, 864460682777, 2512115979800, 7304240074858
Offset: 1
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from T. D. Noe)
- 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
Programs
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Maple
# First construct T(x), the g.f. for A000081. Then we form A005200 = s and its g.f. A as follows: s := [ 1,2 ]; A := series(add(s[ i ]*x^i,i=1..2),x,3); G := series(subs(x=x^2,A),x,3); for n from 3 to 30 do t1 := coeff(T,x,n)+add( coeff(T,x,i)*s[ n-i ],i=1..n-1)-add(coeff(T,x,i)*coeff(G,x,n-i),i=1..n-1); s := [ op(s),t1 ]; A := series(A+t1*x^n,x,n+1); G := series(subs(x=x^2,A),x,n+1); od: s; A; # second Maple program: with(numtheory): b:= proc(n) option remember; local d, j; if n<1 then 0 elif n=1 then 1 else add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1)/ (n-1) fi end: a:= proc(n) option remember; b(n) +add((b(n-i) -b(n-2*i)) *a(i), i=0..n-1) end: seq(a(n), n=1..100); # Alois P. Heinz, Sep 16 2008
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Mathematica
terms = 30; (* T = g.f. of A000081 *) T[x_] = 0; Do[T[x_] = x*Exp[Sum[ T[x^k]/k, {k, 1, terms}]] + O[x]^(terms+1) // Normal, terms+1]; A[] = 0; Do[A[x] = T[x]*(1 + A[x] - A[x^2]) + O[x]^(terms+1) // Normal, terms+1]; Drop[CoefficientList[A[x], x] , 1] (* Jean-François Alcover, Sep 30 2011, updated Jan 11 2018 *) b[n_] := b[n] = Module[{d, j}, If[n<1, 0, If[n == 1, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)]]]; a[n_] := a[n] = b[n] + Sum[ (b[n-i] - b[n-2*i])*a[i], {i, 0, n-1}]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)
Formula
G.f. satisfies A(x)=T(x)[ 1+A(x)-A(x^2) ], where T(x)=x+x^2+2*x^3+... is g.f. for A000081.
Comments