OFFSET
1,3
COMMENTS
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
REFERENCES
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).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..385 (terms 1..100 from T. D. Noe)
H. Bergeron, E. M. F. Curado, J. P. Gazeau and L. M. C. S. Rodrigues, A note about combinatorial sequences and Incomplete Gamma function, arXiv preprint arXiv: 1309.6910 [math.CO], 2013.
J. Riordan, Letter to N. J. A. Sloane, Aug. 1970
J. Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.
FORMULA
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-1) = A000435(n). - M. F. Hasler, Dec 10 2018
E.g.f.: x*diff(A000169(x),x)^2. - Vladimir Kruchinin, Jun 07 2020
a(n) = (n-2)! * Sum_{k=0..n-2} n^k/k! for n > 1. - Jianing Song, Aug 08 2022
MAPLE
seq(simplify(GAMMA(n-1, n)*exp(n)), n=2..20); # Vladeta Jovovic, Jul 21 2005
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved