%I A007562 M0773
%S A007562 1,1,1,2,3,6,10,20,36,72,137,275,541,1098,2208,4521,9240,19084,39451,
%T A007562 82113,171240,358794,753460,1587740,3353192,7100909,15067924,32044456,
%U A007562 68272854,145730675,311575140,667221030,1430892924,3072925944
%N A007562 Number of planted trees where non-root, non-leaf nodes an even distance 
               from root are of degree 2.
%C A007562 There is no planted tree on one node by definition.
%C A007562 G.f. = x+x^2/(Product_{k>0}(1-x^k)^a(k)). - Michael Somos, Oct 06 2003
%D A007562 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A007562 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/
               0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 
               226-228 (1995), 57-72; erratum 320 (2000), 210.
%H A007562 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A007562 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%F A007562 Shifts left 2 places under Euler transform.
%p A007562 with (numtheory): etr:= proc(p) local b; b:= proc(n) option remember; 
               if n=0 then 1 else (add (d*p(d), d=divisors(n)) +add (add(d*p(d), 
               d=divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): a:= 
               n-> if n<=1 then n else b(n-2) fi: seq (a(n), n=1..34); [From Alois 
               P. Heinz (heinz(AT)hs-heilbronn.de), Sep 06 2008]
%o A007562 (PARI) a(n)=local(A); if(n<2,n>0,A=x/(1-x)+O(x^n); for(k=2,n-2,A/=(1-x^k+O(x^n))^polcoeff(A,
               k-1)); polcoeff(A,n-1))
%Y A007562 Sequence in context: A120421 A005418 A002215 this_sequence A008929 A066062 
               A164047
%Y A007562 Adjacent sequences: A007559 A007560 A007561 this_sequence A007563 A007564 
               A007565
%K A007562 nonn,nice,eigen
%O A007562 1,4
%A A007562 N. J. A. Sloane (njas(AT)research.att.com).
%E A007562 Better description from Christian G. Bower (bowerc(AT)usa.net), May 15 
               1998.