Search: id:A005200
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%I A005200 M1247
%S A005200 1,2,4,11,28,78,213,598,1670,4723,13356,37986,108193,309169,884923,
%T A005200 2538369,7292170,20982220,60451567,174385063,503600439,1455827279,
%U A005200 4212464112,12199373350,35357580112,102552754000,297651592188,864460682777
%N A005200 Total number of fixed points in rooted trees with n nodes.
%D A005200 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005200 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 A005200 T. D. Noe, <a href="b005200.txt">Table of n, a(n) for n=1..200</a>
%H A005200 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A005200 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%F A005200 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.
%p A005200 # First construct T(x), the g.f. for A000081. Then we form A005200 = 
               s and its g.f. A as follows:
%p A005200 s := [ 1,2 ]; A := series(add(s[ i ]*x^i,i=1..2),x,3); G := series(subs(x=x^2,
               A),x,3);
%p A005200 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;
%p A005200 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); [From Alois 
               P. Heinz (heinz(AT)hs-heilbronn.de), Sep 16 2008]
%Y A005200 Cf. A000081, A005201, A000055.
%Y A005200 Sequence in context: A007048 A148132 A032101 this_sequence A148133 A148134 
               A151425
%Y A005200 Adjacent sequences: A005197 A005198 A005199 this_sequence A005201 A005202 
               A005203
%K A005200 nonn,easy,nice
%O A005200 1,2
%A A005200 N. J. A. Sloane (njas(AT)research.att.com).

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