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A001863
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Normalized total height of rooted trees with n nodes.
(Formerly M3614 N1466)
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9
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0, 1, 4, 26, 236, 2760, 39572, 672592, 13227804, 295579520, 7398318500, 205075286784, 6236796259916, 206489747516416, 7393749269685300, 284714599444490240, 11733037015160276348, 515240326393584058368, 24019843795708471562564, 1184776250223810469888000
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OFFSET
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1,3
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COMMENTS
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a(n) is the number of partial functions f from [n-1] into [n-1] such that f^k(1) is undefined for some k>=1. - Geoffrey Critzer, Mar 05 2022
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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E.g.f.: -exp(1)*x*(Ei(-1-LambertW(-x))-Ei(-1)) - LambertW(-x) + log(1+LambertW(-x)). - Vladeta Jovovic, Sep 29 2003
a(n) = (n-2)! * Sum_{k=0..n-2} n^k/k! for n > 1. - Jianing Song, Aug 08 2022
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MAPLE
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A001863 := n->add((n-2)!*n^k/k!, k=0..n-2); # for n>1. Equals A001864(n)/(n^2-n)
seq(simplify(GAMMA(n-1, n)*exp(n)), n=2..20); # Vladeta Jovovic, Jul 21 2005
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MATHEMATICA
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a[n_] := Sum[(n-2)!*n^k/k!, {k, 0, n-2}]; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Oct 09 2012, from Maple *)
Table[Sum[(n-2)! n^k/k!, {k, 0, n-2}], {n, 30}] (* Harvey P. Dale, Jun 19 2016 *)
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PROG
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(PARI) apply( A001863(n)=sum(k=0, n-2, (n-2)!/k!*n^k), [1..20]) \\ This defines the function A001863; apply(...) provides a check and illustration. - G. C. Greubel, Nov 14 2017, edited by M. F. Hasler, Dec 09 2018
(Magma) [0] cat [&+[Factorial(n-2)*n^k div Factorial(k): k in [0..n-2]]: n in [2..24]]; // Vincenzo Librandi, Dec 10 2018
(Python)
from math import comb
def A001863(n): return 0 if n<2 else ((sum(comb(n, k)*(n-k)**(n-k)*k**k for k in range(1, (n+1>>1)))<<1) + (0 if n&1 else comb(n, m:=n>>1)*m**n))//n//(n-1) # Chai Wah Wu, Apr 25-26 2023
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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