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A038156 a(n) = n! * Sum_{k=1..n-1} 1/k!.

%I A038156 #92 Aug 22 2025 19:55:20
%S A038156 0,0,2,9,40,205,1236,8659,69280,623529,6235300,68588311,823059744,
%T A038156 10699776685,149796873604,2246953104075,35951249665216,
%U A038156 611171244308689,11001082397556420,209020565553571999,4180411311071440000,87788637532500240021,1931350025715005280484
%N A038156 a(n) = n! * Sum_{k=1..n-1} 1/k!.
%C A038156 Related to number of operations of addition and multiplication to evaluate a determinant of order n by cofactor expansion - see A026243.
%C A038156 Also number of operations needed to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of comparisons required to find j in step L2 (see answer to exercise 5). - _Hugo Pfoertner_, Jan 24 2003
%C A038156 For n>1, the number of possible ballots where there are n candidates and voters may identify their top m most preferred ones, where 0 < m < n. - _Shaye Horwitz_, Jun 28 2011
%C A038156 For n > 1, a(n) is the expected number of comparisons required to sort a random list of n distinct elements using the "bogosort" algorithm. - _Andrew Slattery_, Jun 02 2022
%C A038156 The number of permutations of all proper nonempty subsets of an n element set. - _P. Christopher Staecker_, May 09 2024
%D A038156 D. E. Knuth: The Art of Computer Programming, Volume 4, Fascicle 2. Generating All Tuples and Permutations, Addison-Wesley, 2005.
%H A038156 Georg Fischer, <a href="/A038156/b038156.txt">Table of n, a(n) for n = 0..200</a> [first 28 terms from _Vincenzo Librandi_]
%H A038156 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=836">Encyclopedia of Combinatorial Structures 836</a>
%H A038156 G. A. Kamel, <a href="http://www.aascit.org/journal/archive2?journalId=928&amp;paperId=2310">Partial Chain Topologies on Finite Sets</a>, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
%H A038156 Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/lpure.txt">FORTRAN implementation of Knuth's Algorithm L for lexicographic permutation generation</a>.
%H A038156 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bogosort">Bogosort</a>
%H A038156 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A038156 a(n) = floor((e-1)*n!) - 1.
%F A038156 a(0) = a(1) = 0, a(n) = n*(a(n-1) + 1) for n>1. - _Philippe Deléham_, Oct 16 2009
%F A038156 E.g.f.: (exp(x) - 1)*x/(1 - x). - _Ilya Gutkovskiy_, Jan 26 2017
%F A038156 a(n) = A002627(n)-1, n>=1. - _R. J. Mathar_, Jan 03 2018
%F A038156 a(n) = A000522(n)-n!-1, n>=1. - _P. Christopher Staecker_, May 09 2024
%e A038156 a(2) = floor((2.718... - 1)*2) - 1 = 3 - 1 = 2,
%e A038156 a(3) = floor((2.718... - 1)*6) - 1 = 10 - 1 = 9.
%p A038156 a:= proc(n) option remember;
%p A038156       `if`(n<2, 0, a(n-1)*n+n)
%p A038156     end:
%p A038156 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 11 2020
%t A038156 a=1; Join[{0},Table[a=(a-1)*(n+1);Abs[a],{n,0,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Nov 20 2009; 0 prefixed by _Georg Fischer Apr 11 2020 *)
%t A038156 Join[{0},FoldList[#1*#2 + #2 + #1 + 1 &, 0, Range@ 20]] (* _Robert G. Wilson v_, Feb 21 2015 *)
%o A038156 (PARI) a(n)=floor((exp(1)-1)*n!-1) \\ _Charles R Greathouse IV_, Jun 29 2011
%o A038156 (PARI) a(n)=(expm1(1)*n!-1)\1 \\ _Charles R Greathouse IV_, Jan 28 2014
%Y A038156 Cf. A000522, A007526, A038155, A056542, A079884, A079750.
%Y A038156 Row sums of A119741.
%K A038156 nonn,easy,changed
%O A038156 0,3
%A A038156 _N. J. A. Sloane_
%E A038156 a(28) ff. corrected by _Georg Fischer_, Apr 11 2020