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%I #9 May 21 2018 15:09:59
%S 1,1,1,2,5,13,32,80,200,511,1323,3471,9183,24491,65715,177363,481135,
%T 1311340,3589023,9860254,27181835,75165194,208439742,579522977,
%U 1615093755,4511122964,12625881944,35405197065,99459085125,279861792874,788712430532,2226015529592
%N Number of unlabeled rooted trees with n vertices such that every branch of the root has a different number of leaves.
%H Andrew Howroyd, <a href="/A298535/b298535.txt">Table of n, a(n) for n = 1..500</a>
%t rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];
%t Table[Length[Select[rut[n],UnsameQ@@(Count[#,{},{0,Infinity}]&/@#)&]],{n,15}]
%o (PARI) \\ here R is A055277 as vector of polynomials
%o 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)};
%o 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
%Y Cf. A000081, A003238, A004111, A032305, A289079, A290689, A291443, A297791, A298422, A298533, A298536.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jan 20 2018
%E Terms a(19) and beyond from _Andrew Howroyd_, May 20 2018
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