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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 31 2018
EXTENSIONS
Terms a(26) and beyond from Andrew Howroyd, Aug 28 2018
STATUS
approved