A298535 Number of unlabeled rooted trees with n vertices such that every branch of the root has a different number of leaves.

1, 1, 1, 2, 5, 13, 32, 80, 200, 511, 1323, 3471, 9183, 24491, 65715, 177363, 481135, 1311340, 3589023, 9860254, 27181835, 75165194, 208439742, 579522977, 1615093755, 4511122964, 12625881944, 35405197065, 99459085125, 279861792874, 788712430532, 2226015529592

Offset: 1

Author

Gus Wiseman, Jan 20 2018

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A000081, A003238, A004111, A032305, A289079, A290689, A291443, A297791, A298422, A298533, A298536.

Programs

Mathematica

rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];
Table[Length[Select[rut[n],UnsameQ@@(Count[#,{},{0,Infinity}]&/@#)&]],{n,15}]

PARI

\\ here R is A055277 as vector of polynomials
R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1  + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
seq(n) = {my(M=Mat(apply(p->Colrev(p,n), R(n-1)))); Vec(prod(i=2, #M, 1 + x*Ser(M[i,])))} \\ Andrew Howroyd, May 20 2018

Extensions

Terms a(19) and beyond from Andrew Howroyd, May 20 2018