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%I M3614 N1466 #85 Apr 26 2023 18:33:39
%S 0,1,4,26,236,2760,39572,672592,13227804,295579520,7398318500,
%T 205075286784,6236796259916,206489747516416,7393749269685300,
%U 284714599444490240,11733037015160276348,515240326393584058368,24019843795708471562564,1184776250223810469888000
%N Normalized total height of rooted trees with n nodes.
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A001863/b001863.txt">Table of n, a(n) for n = 1..385</a> (terms 1..100 from T. D. Noe)
%H H. Bergeron, E. M. F. Curado, J. P. Gazeau and L. M. C. S. Rodrigues, <a href="http://arxiv.org/abs/1309.6910">A note about combinatorial sequences and Incomplete Gamma function</a>, arXiv preprint arXiv: 1309.6910 [math.CO], 2013.
%H J. Riordan, <a href="/A001863/a001863.pdf">Letter to N. J. A. Sloane, Aug. 1970</a>
%H J. Riordan and N. J. A. Sloane, <a href="http://dx.doi.org/10.1017/S1446788700007527">Enumeration of rooted trees by total height</a>, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F E.g.f.: -exp(1)*x*(Ei(-1-LambertW(-x))-Ei(-1)) - LambertW(-x) + log(1+LambertW(-x)). - _Vladeta Jovovic_, Sep 29 2003
%F a(n)*(n-1) = A000435(n). - _M. F. Hasler_, Dec 10 2018
%F E.g.f.: x*diff(A000169(x),x)^2. - _Vladimir Kruchinin_, Jun 07 2020
%F a(n) = (n-2)! * Sum_{k=0..n-2} n^k/k! for n > 1. - _Jianing Song_, Aug 08 2022
%p A001863 := n->add((n-2)!*n^k/k!, k=0..n-2); # for n>1. Equals A001864(n)/(n^2-n)
%p seq(simplify(GAMMA(n-1,n)*exp(n)),n=2..20); # _Vladeta Jovovic_, Jul 21 2005
%t 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 *)
%t Table[Sum[(n-2)! n^k/k!,{k,0,n-2}],{n,30}] (* _Harvey P. Dale_, Jun 19 2016 *)
%o (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
%o (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
%o (Python)
%o from math import comb
%o 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
%Y Cf. A000435, A000169, A001864.
%K nonn,nice,easy
%O 1,3
%A _N. J. A. Sloane_
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