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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 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

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Last modified December 2 07:09 EST 2023. Contains 367510 sequences. (Running on oeis4.)