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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 (list; graph; refs; listen; history; text; internal format)
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..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
Sequence in context: A198077 A208813 A112698 * A344051 A025168 A084358
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 14 2014
STATUS
approved

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Last modified March 28 13:21 EDT 2024. Contains 371254 sequences. (Running on oeis4.)