%I #22 Jun 21 2018 15:14:10
%S 1,1,1,3,5,13,35,95,249,691,2007,5719,16823,49371,146755,438301,
%T 1319343,3981699,12129477,36987253,113456615,348921105,1077206189,
%U 3332120237,10347481901,32183230157,100372658801,313633257399,982232930081,3080379360481,9681909324247
%N Number of planted planar trees (n+1 nodes) where any 2 subtrees extending from the same node have a different number of nodes.
%C Number of compositions of n-1 into distinct parts if there are a(i) kinds of part i. a(6) = 13: 5, 5', 5'', 5''', 5'''', 41, 4'1, 4''1, 14, 14', 14'', 32, 23.
%H Alois P. Heinz, <a href="/A032009/b032009.txt">Table of n, a(n) for n = 1..1938</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F Shifts left under "AFK" (ordered, size, unlabeled) transform
%p b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p b(n, i-1, p) +`if`(i>n, 0, a(i)*b(n-i, i-1, p+1))))
%p end:
%p a:= n-> b(n-1$2, 0):
%p seq(a(n), n=1..40); # _Alois P. Heinz_, Sep 05 2015
%t b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, b[n, i-1, p] + If[ i>n, 0, a[i]*b[n-i, i-1, p+1]]]]; a[n_] := b[n-1, n-1, 0]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Feb 24 2016, after _Alois P. Heinz_ *)
%o (PARI)
%o AFK(v)={apply(p->subst(serlaplace(y^0*p), y, 1), Vec(prod(k=1, #v, 1 + v[k]*x^k*y + O(x*x^#v))))}
%o seq(n)={my(v=[1]); for(i=1, n, v=AFK(v)); v} \\ _Andrew Howroyd_, Jun 21 2018
%Y Cf. A261894.
%K nonn,eigen
%O 1,4
%A _Christian G. Bower_
%E More terms from _Alois P. Heinz_, Sep 05 2015