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A005201
Total number of fixed points in trees with n nodes.
(Formerly M3803)
4
1, 0, 1, 1, 5, 10, 31, 72, 201, 509, 1374, 3587, 9647, 25686, 69348, 187052, 508480, 1384959, 3791466, 10407842, 28677319, 79231664, 219557624, 609922977, 1698526750, 4740469708, 13258136509, 37151664771, 104294992317, 293279485007
OFFSET
1,5
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, The probability that a point of a tree is fixed, Math. Proc. Camb. Phil. Soc. 85(1979) 407-415.
FORMULA
G.f. satisfies A(x) = T(x)*(1-F(x^2))-F(x^2), where T(x) = x + x^2 + 2*x^3 + ... is g.f. for A000081, F(x) = x + 2*x^2 + 4*x^3 + 11*x^4 + ... is the g.f. for A005200.
MAPLE
# First form T(x) = g.f. for A000081 and F(x) = g.f. for A005200. Then:
t1 := subs(x=x^2, F); series(T*(1-t1)-t1, x, 31);
# second Maple program:
with(numtheory): t:= proc(n) option remember; local d, j; if n<1 then 0 elif n=1 then 1 else add(add(d*t(d), d=divisors(j)) *t(n-j), j=1..n-1)/ (n-1) fi end: f:= proc(n) option remember; t(n) +add((t(n-i) -t(n-2*i)) *f(i), i=0..n-1) end: t1 := n-> `if`(type(n, odd), 0, f(n/2)): a:= proc(n) t(n) -add(t(n-i) *t1(i), i=0..n) -t1(n) end: seq(a(n), n=1..50); # Alois P. Heinz, Sep 17 2008
MATHEMATICA
t[n_] := t[n] = If[n<1, 0, If[n == 1, 1, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]]; f[n_] := f[n] = t[n]+Sum[(t[n-i]-t[n-2*i])*f[i], {i, 0, n-1}]; t1[n_] := If[OddQ[n], 0, f[n/2]]; a[n_] := t[n]-Sum[t[n-i]*t1[i], {i, 0, n}]-t1[n]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,easy,nice
STATUS
approved