[seqfan] Re: Number of Trees

N. J. A. Sloane njas at research.att.com
Fri Mar 6 23:16:56 CET 2009

Jim,  the reason there is maple code for A81 given in A55 is
that the first Maple line in A55 expresses the gf in terms
of the gf for A81


%F A000055 G.f.: A(x) = 1 + T(x)-T^2(x)/2+T(x^2)/2, where T(x) = x + x^2 + 2*x^3 + ... is g.f. \
for A000081

and then:

%p A000055 G000055 := series(1+G000081-G000081^2/2+subs(x=x^2,G000081)/2,x,31); A000055 := n->c\
oeff(G000055,x,n); # where G000081 is g.f. for A000081 starting with n=1 term
%p A000055 b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n\
-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) opt\
ion remember; unapply (add (b(k)*x^k, k=1..n),x) end: a:= n-> coeff (series (1+ B(n)(x)- (B(n)(\
x))^2/2+ B(n)(x^2)/2, x=0, n+1),x,n): seq (a(n), n=0..32); [From Alois P. Heinz (heinz(AT)hs-he\
ilbronn.de), Aug 21 2008]

b = A81, a = A55


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