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A317580 Number of unlabeled rooted identity trees with n nodes and a distinguished leaf. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 17 07:39 EDT 2021. Contains 348048 sequences. (Running on oeis4.)