A309352 Number of free trees of n vertices whose automorphisms are a 2-group.

1, 1, 1, 1, 1, 2, 4, 6, 13, 26, 56, 122, 278, 634, 1494, 3540, 8542, 20774, 51116, 126648, 316452, 795510, 2012476, 5117613, 13079677, 33576706, 86555074, 223965633, 581573118, 1515084771, 3959038337, 10374543765, 27258298145

Offset: 0

Author

Kevin Ryde, Jul 24 2019

Keywords

• nonn

Comments

A 2-group is a group with order some power 2^k. In free tree automorphisms, this means at a given vertex an adjacent subtree can appear there twice, but not three or more times.

Free trees are counted from rooted trees in the usual way by rooting at a centroid. This is in the generating function of A000055 (all free) from A000081 (all rooted). From all rooted trees, subtract the unbalanced where root not centroid, and allow bicentroidals with same halves. a(n) follows this way from A248869 rooted 2-groups. With root as centroid, the free automorphisms are all and only the rooted automorphisms, except swap halves of a bi-centroidal. Such a swap can be part of a 2-group, so same halves allowed.

Examples

			a(4)=1 is path-4 having automorphism group S2 (reverse the path), and excludes star-4 which is S3 order 6 (permute the leaves).  a(5)=2 excludes star-5 which is S4 on the leaves.
		

Links

• Kevin Ryde, Table of n, a(n) for n = 0..2213

Crossrefs

Cf. A000055, A000220, A248869.

Programs

Maple

h:= proc(n, m, t) option remember; `if`(m=0, binomial(n+t, t),
      `if`(n=0, 0, add(h(n-1, m-j, t+1), j=1..min(2, m))))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*h(g(i), j, 0), j=0..n/i)))
    end:
g:= n-> `if`(n<2, n, b(n-1$2)):
a:= n-> `if`(n=0, 1, g(n)-add(g(j)*g(n-j), j=0..n/2)+
        `if`(n::even, (t-> t*(t+1)/2)(g(n/2)), 0)):
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 01 2019

Mathematica

h[n_, m_, t_] := h[n, m, t] = If[m == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n-1, m-j, t+1], {j, 1, Min[2, m]}]]];
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[b[n - i j, i-1] h[g[i], j, 0], {j, 0, n/i}]]];
g[n_] := If[n < 2, n, b[n-1, n-1]];
a[n_] := If[n == 0, 1, g[n] - Sum[g[j] g[n-j], {j, 0, n/2}] + If[EvenQ[n], #(#+1)/2&[g[n/2]], 0]];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz *)

Formula

a(n) = A248869(n) - (Sum_{i=1..floor(n/2)} A248869(i)*A248869(n-i)) + (binomial(A248869(n/2)+1,2) if n even), for n>=1.

G.f.: g(x) = 1 + T(x) - T^2(x)/2 + T(x^2)/2, where T(x) = x + x^2 + 2*x^3 + 3*x^4 + ... is the g.f. for A248869.