%I #10 Aug 28 2018 23:37:57
%S 1,1,1,3,5,12,28,66,153,367,880,2121,5127,12441,30248,73746,180077,
%T 440571,1079438,2648511,6506170,16001256,39393173,97074140,239419963,
%U 590972968,1459808862,3608483107,8925476591,22090139751,54702648393,135533335933,335967782916
%N Number of unlabeled rooted identity trees with n nodes and a distinguished leaf.
%C Total number of leaves in all rooted identity trees with n nodes. - _Andrew Howroyd_, Aug 28 2018
%H Andrew Howroyd, <a href="/A317580/b317580.txt">Table of n, a(n) for n = 1..200</a>
%F a(n) = Sum_{k=1, n} k*A055327(n, k). - _Andrew Howroyd_, Aug 28 2018
%e The a(6) = 12 rooted identity trees with a distinguished leaf:
%e (((((O))))),
%e (((O(o)))), (((o(O)))),
%e ((O((o)))), ((o((O)))),
%e (O(((o)))), (o(((O)))),
%e ((O)((o))), ((o)((O))),
%e (O(o(o))), (o(O(o))), (o(o(O))).
%t urit[n_]:=Join@@Table[Select[Union[Sort/@Tuples[urit/@ptn]],UnsameQ@@#&],{ptn,IntegerPartitions[n-1]}];
%t Table[Sum[Length[Flatten[{t/.{}->1}]],{t,urit[n]}],{n,10}]
%o (PARI) WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
%o seq(n)={my(v=[y]); for(n=2, n, v=concat([y], WeighMT(v))); apply(p -> subst(deriv(p), y, 1), v)} \\ _Andrew Howroyd_, Aug 28 2018
%Y Cf. A000081, A001678, A003227, A003238, A004111, A038046, A055327, A067824, A301342, A316784.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jul 31 2018
%E Terms a(26) and beyond from _Andrew Howroyd_, Aug 28 2018