A317580 Number of unlabeled rooted identity trees with n nodes and a distinguished leaf.

1, 1, 1, 3, 5, 12, 28, 66, 153, 367, 880, 2121, 5127, 12441, 30248, 73746, 180077, 440571, 1079438, 2648511, 6506170, 16001256, 39393173, 97074140, 239419963, 590972968, 1459808862, 3608483107, 8925476591, 22090139751, 54702648393, 135533335933, 335967782916

Offset: 1

Author

Gus Wiseman, Jul 31 2018

Keywords

• nonn

Comments

Total number of leaves in all rooted identity trees with n nodes. - Andrew Howroyd, Aug 28 2018

Examples

			The a(6) = 12 rooted identity trees with a distinguished leaf:
(((((O))))),
(((O(o)))), (((o(O)))),
((O((o)))), ((o((O)))),
(O(((o)))), (o(((O)))),
((O)((o))), ((o)((O))),
(O(o(o))), (o(O(o))), (o(o(O))).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000081, A001678, A003227, A003238, A004111, A038046, A055327, A067824, A301342, A316784.

Programs

Mathematica

urit[n_]:=Join@@Table[Select[Union[Sort/@Tuples[urit/@ptn]],UnsameQ@@#&],{ptn,IntegerPartitions[n-1]}];
Table[Sum[Length[Flatten[{t/.{}->1}]],{t,urit[n]}],{n,10}]

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)}
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

Formula

a(n) = Sum_{k=1, n} k*A055327(n, k). - Andrew Howroyd, Aug 28 2018

Extensions

Terms a(26) and beyond from Andrew Howroyd, Aug 28 2018