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

1,4

Comments

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

Links

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

Formula

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

Example

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))).

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}]

Prog

(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

Crossrefs

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

Sequence in context: A291035 A005913 A056690 * A066951 A295065 A046091

Adjacent sequences: A317577 A317578 A317579 * A317581 A317582 A317583

Keyword

nonn

Author

Gus Wiseman, Jul 31 2018

Extensions

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

Status

approved