

A234953


Normalized total height of all rooted trees on n labeled nodes.


7



0, 1, 5, 37, 357, 4351, 64243, 1115899, 22316409, 505378207, 12789077631, 357769603027, 10965667062133, 365497351868767, 13163965052815515, 509522144541045811, 21093278144993719665, 930067462093579181119, 43518024090910884374263, 2153670733766937656155699
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OFFSET

1,3


COMMENTS

Equals A001854(n)/n. That is, similar to A001854, except here the root always has the fixed label 1.
This was in one of my thesis notebooks from 1964 (see the scans in A000435), but because it wasn't of central importance it was never added to the OEIS.


LINKS



FORMULA

a(n) = Sum_{k=1..n1} k*A034855(n,k)/n = Sum_{k=1..n1} k*A235595(n,k).


MATHEMATICA

gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k1]]; a[n_, k_] := n!*Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; a[n_] := Sum[k*(a[n, k]  a[n, k1]), {k, 1, n1}]/n; Array[a, 20] (* JeanFrançois Alcover, Mar 18 2014, after Alois P. Heinz *)


PROG

(Python)
from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, h): return 1 if min(n, h)==0 else sum([binomial(n  1, j  1)*j*b(j  1, h  1)*b(n  j, h) for j in range(1, n + 1)])
def T(n, k): return b(n  1, k  1)  b(n  1, k  2)
def a(n): return sum([k*T(n, k) for k in range(1, n)])


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



