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%I #28 Apr 21 2021 04:34:27
%S 0,1,5,37,357,4351,64243,1115899,22316409,505378207,12789077631,
%T 357769603027,10965667062133,365497351868767,13163965052815515,
%U 509522144541045811,21093278144993719665,930067462093579181119,43518024090910884374263,2153670733766937656155699
%N Normalized total height of all rooted trees on n labeled nodes.
%C Equals A001854(n)/n. That is, similar to A001854, except here the root always has the fixed label 1.
%C 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.
%H Alois P. Heinz, <a href="/A234953/b234953.txt">Table of n, a(n) for n = 1..387</a>
%F a(n) = Sum_{k=1..n-1} k*A034855(n,k)/n = Sum_{k=1..n-1} k*A235595(n,k).
%t 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_ *)
%o (Python)
%o from sympy import binomial
%o from sympy.core.cache import cacheit
%o @cacheit
%o 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)])
%o def T(n, k): return b(n - 1, k - 1) - b(n - 1, k - 2)
%o def a(n): return sum([k*T(n, k) for k in range(1, n)])
%o print([a(n) for n in range(1, 31)]) # _Indranil Ghosh_, Aug 26 2017
%Y Cf. A001854, A034855, A235595, A236396.
%K nonn
%O 1,3
%A _N. J. A. Sloane_, Jan 14 2014
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