[seqfan] Re: Number of Trees

[N. J. A. Sloane]

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

Look:

%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

Neil