%I M2818 #188 Apr 15 2024 12:52:02

%S 0,1,3,9,33,153,873,5913,46233,409113,4037913,43954713,522956313,

%T 6749977113,93928268313,1401602636313,22324392524313,378011820620313,

%U 6780385526348313,128425485935180313,2561327494111820313,53652269665821260313,1177652997443428940313

%N a(n) = Sum_{k=1..n} k!.

%C Equals row sums of triangle A143122 starting (1, 3, 9, 33, ...). - _Gary W. Adamson_, Jul 26 2008

%C a(n) for n>=4 is never a perfect square. - _Alexander R. Povolotsky_, Oct 16 2008

%C Number of cycles that can be written in the form (j,j+1,j+2,...), in all permutations of {1,2,...,n}. Example: a(3)=9 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), (132) we have 3+2+2+1+1+0=9 such cycles. - _Emeric Deutsch_, Jul 14 2009

%C Conjectured to be the length of the shortest word over {1,...,n} that contains each of the n! permutations as a factor (cf. A180632) [see Johnston]. - _N. J. A. Sloane_, May 25 2013

%C The above conjecture has been disproven for n>=6. See A180632 and the Houston 2014 reference. - _Dmitry Kamenetsky_, Mar 07 2016

%C a(n) is also the number of compositions of n if cardinal values do not matter but ordinal rankings do. Since cardinal values do not matter, a sequence of k summands summing to n can be represented as (s(1),...,s(k)), where the s's are positive integers and the numbers in parentheses are the initial ordinal rankings. The number of compositions of these summands are equal to k!, with k ranging from 1 to n. - _Gregory L. Simay_, Jul 31 2016

%C When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the left. Compare array A211370 for circular shifts to the left in a broader sense. Compare sequence A001563 for circular shifts to the right. - _Tilman Piesk_, Apr 29 2017

%C Since a(n) = (1!+2!+3!+...+n!) = 3(1+3!/3+4!/3+...+n!/3) is a multiple of 3 for n>2, the only prime in this sequence is a(2) = 3. - _Eric W. Weisstein_, Jul 15 2017

%C Generalization of 2nd comment: a(n) for n>=4 is never a perfect power (A007916) (Chentzov link). - _Bernard Schott_, Jan 26 2023

%D R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Section B44, Springer 2010.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Carauleanu Marc, <a href="/A007489/b007489.txt">Table of n, a(n) for n = 0..212</a> (first 100 terms from T. D. Noe)

%H N. N. Chentzov, D. O. Shklarsky, and I. M. Yaglom, <a href="https://archive.org/details/ussr_olympiad_problem_book/page/n106/mode/1up">The USSR Olympiad Problem Book, Selected Problems and Theorems of Elementary Mathematics</a>, problem 115, pp. 28 and 201-202, Dover publications, Inc., New York, 1993.

%H R. K. Guy, <a href="/A005728/a005728.pdf">Letter to N. J. A. Sloane, 1987</a>

%H Robin Houston, <a href="http://arxiv.org/abs/1408.5108">Tackling the Minimal Superpermutation Problem</a>, arXiv:1408.5108 [math.CO], 2014.

%H Nathaniel Johnston, <a href="http://www.njohnston.ca/2013/04/the-minimal-superpermutation-problem/">The minimal superpermutation problem</a> (2013)

%H Nathaniel Johnston, <a href="http://dx.doi.org/10.1016/j.disc.2013.03.024">Non-uniqueness of minimal superpermutations</a>, Discrete Math. 313 (2013), no. 14, 1553--1557. MR3047396

%H S. Legendre and P. Paclet, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Legendre/legendre5.html">On the Permutations Generated by Cyclic Shift </a>, J. Int. Seq. 14 (2011) # 11.3.2.

%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha132.htm">Factorizations of many number sequences</a>

%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>

%H Alexsandar Petojevic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Factorial.html">Factorial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeftFactorial.html">Left Factorial</a>

%H G. Xiao, Sigma Server, <a href="http://wims.unice.fr/~wims/en_tool~analysis~sigma.en.html">Operate on "n!"</a>

%H Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 5.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F a(n) = Sum_{k=1..n} P(n, k)/C(n, k). - _Ross La Haye_, Sep 21 2004

%F a(n) = 3*A056199(n) for n>=2. - _Philippe Deléham_, Feb 10 2007

%F a(n) = !(n+1)-1=A003422(n+1)-1. - _Artur Jasinski_, Nov 08 2007 [corrected by _Werner Schulte_, Oct 20 2021]

%F Starting (1, 3, 9, 33, 153, ...), = row sums of triangle A137593 - _Gary W. Adamson_, Jan 28 2008

%F a(n) = a(n-1) + n! for n >= 1. - _Jaroslav Krizek_, Jun 16 2009

%F E.g.f. A(x) satisfies to the differential equation A'(x)=A(x)+x/(1-x)^2+1. - _Vladimir Kruchinin_, Jan 22 2011

%F a(0)=0, a(1)=1, a(n) = (n+1)*a(n-1)-n*a(n-2). - _Sergei N. Gladkovskii_, Jul 05 2012

%F G.f.: W(0)*x/(2-2*x) , where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 19 2013

%F G.f.: x /(1-x)/Q(0),m=+2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - _Sergei N. Gladkovskii_, Sep 24 2013

%F E.g.f.: exp(x-1)*(Ei(1) - Ei(1-x)) - exp(x) + 1/(1 - x), where Ei(x) is the exponential integral. - _Ilya Gutkovskiy_, Nov 27 2016

%F a(n) = sqrt(a(n-1)*a(n+1)-a(n-2)*n*n!), n >= 2. - _Gary Detlefs_, Oct 26 2020

%e a(4) = 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33. - _Michael B. Porter_, Aug 03 2016

%p A007489 := proc(n) local i; add(i!,i=1..n); end proc;

%t FoldList[Plus, 0, (Range@ 21)! ] (* _Robert G. Wilson v_, Sep 21 2007 *)

%t Table[Sum[i!, {i, 1, n}], {n, 0, 21}] (* _Zerinvary Lajos_, Jul 12 2009 *)

%t Accumulate[Range[50]!] (* _Harvey P. Dale_, Apr 30 2011 *)

%t Table[Plus@@(Range[n]!), {n, 20}] (* _Alonso del Arte_, Jul 18 2011 *)

%o (PARI) a(n)=sum(k=1,n,k!) \\ _Charles R Greathouse IV_, Jul 25 2011

%o (Haskell)

%o a007489 n = a007489_list !! n

%o a007489_list = scanl (+) 0 $ tail a000142_list

%o -- _Reinhard Zumkeller_, Aug 29 2014

%o (Magma) [0] cat [&+[Factorial(i): i in [1..n]]: n in [1..25]]; // _Vincenzo Librandi_, Sep 02 2016

%o (GAP) List([1..20],n->Sum([1..n],Factorial)); # _Muniru A Asiru_, Jan 31 2018

%Y Equals A003422(n+1) - 1.

%Y Cf. A000142, A000670, A001597, A007916, A137593, A143122, A161128, A180632.

%Y Column k=0 of A120695.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_, _Robert G. Wilson v_