A234953 Normalized total height of all rooted trees on n labeled nodes.

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

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

Alois P. Heinz, Table of n, a(n) for n = 1..387

Formula

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

Mathematica

gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k-1]]; a[n_, k_] := n!*Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; a[n_] := Sum[k*(a[n, k] - a[n, k-1]), {k, 1, n-1}]/n; Array[a, 20] (* Jean-François 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)])
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Aug 26 2017

Crossrefs

Cf. A001854, A034855, A235595, A236396.

Sequence in context: A198077 A208813 A112698 * A344051 A025168 A084358

Adjacent sequences: A234950 A234951 A234952 * A234954 A234955 A234956

Keyword

nonn

Author

N. J. A. Sloane, Jan 14 2014

Status

approved