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%I M2858 N1149 #170 Apr 15 2024 12:59:44
%S 0,1,3,10,41,206,1237,8660,69281,623530,6235301,68588312,823059745,
%T 10699776686,149796873605,2246953104076,35951249665217,
%U 611171244308690,11001082397556421,209020565553572000,4180411311071440001,87788637532500240022
%N a(n) = n*a(n-1) + 1, a(0) = 0.
%C This sequence shares divisibility properties with A000522; each of the primes in A072456 divide only a finite number of terms of this sequence. - _T. D. Noe_, Jul 07 2005
%C Sum of the lengths of the first runs in all permutations of [n]. Example: a(3)=10 because the lengths of the first runs in the permutation (123),(13)2,(3)12,(2)13,(23)1 and (3)21 are 3,2,1,1,2 and 1, respectively (first runs are enclosed between parentheses). Number of cells in the last columns of all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. a(n) = Sum_{k=1..n} k*A092582(n,k). - _Emeric Deutsch_, Aug 16 2006
%C Starting with offset 1 = eigensequence of an infinite lower triangular matrix with (1, 2, 3, ...) as the right border, (1, 1, 1, ...) as the left border, and the rest zeros. - _Gary W. Adamson_, Apr 27 2009
%C Sums of rows of the triangle in A173333, n > 0. - _Reinhard Zumkeller_, Feb 19 2010
%C if s(n) is a sequence defined as s(0) = x, s(n) = n*s(n-1)+k, n > 0 then s(n) = n!*x + a(n)*k. - _Gary Detlefs_, Feb 20 2010
%C Number of arrangements of proper subsets of n distinct objects, i.e., arrangements which are not permutations (where the empty set is considered a proper subset of any nonempty set); see example. - _Daniel Forgues_, Apr 23 2011
%C For n >= 0, A002627(n+1) is the sequence of sums of Pascal-like triangle with one side 1,1,..., and the other side A000522. - _Vladimir Shevelev_, Feb 06 2012
%C a(n) = q(n,1) for n >= 1, where the polynomials q are defined at A248669. - _Clark Kimberling_, Oct 11 2014
%C a(n) is the number of quasilinear weak orderings on {1,...,n}. - _J. Devillet_, Dec 22 2017
%D D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.
%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 Seiichi Manyama, <a href="/A002627/b002627.txt">Table of n, a(n) for n = 0..449</a> (terms 0..100 from T. D. Noe)
%H Sanka Balasuriya, Igor E. Shparlinski and Arne Winterhof, <a href="https://doi.org/10.1216/RMJ-2009-39-5-1403">An average bound for character sums with some counter-dependent recurrence sequences</a>, Rocky Mt. J. Math. 39, No. 5, 1403-1409 (2009).
%H E. Barcucci, A. Del Lungo and R. Pinzani, <a href="http://dx.doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.
%H Jonathan Beagley and Lara Pudwell, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Pudwell/pudwell13.html">Colorful Tilings and Permutations</a>, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
%H J. Devillet, <a href="https://arxiv.org/abs/1712.07856">Bisymmetric and quasitrivial operations: characterizations and enumerations</a>, [math.RA] arXiv:1712.07856 (2017).
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=150">Encyclopedia of Combinatorial Structures 150</a>
%H D. Singh, <a href="/A002627/a002627.pdf">The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers</a>, Math. Student, 20 (1952), 66-70. [Annotated scanned copy]
%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.
%F a(n) = n! * Sum_{k=1..n} 1/k!.
%F a(n) = A000522(n) - n!. - _Michael Somos_, Mar 26 1999
%F a(n) = floor( n! * (e-1) ), n >= 1. - _Amarnath Murthy_, Mar 08 2002
%F E.g.f.: (exp(x)-1)/(1-x). - Mario Catalani (mario.catalani(AT)unito.it), Feb 06 2003
%F Binomial transform of A002467. - _Ross La Haye_, Sep 21 2004
%F a(n) = Sum_{j=1..n} (n-j)!*binomial(n,j). - _Zerinvary Lajos_, Jul 31 2006
%F a(n) = 1 + Sum_{k=0..n-1} k*a(k). - _Benoit Cloitre_, Jul 26 2008
%F a(m) = int(((1+s)^m - s^m)*exp(-s), s=0..infinity) = GAMMA(m+1,1) * exp(1) - GAMMA(m+1). - _Stephen Crowley_, Jul 24 2009
%F From _Sergei N. Gladkovskii_, Jul 05 2012: (Start)
%F a(n+1) = A000522(n) + A001339(n) - A000142(n+1);
%F E.g.f.: Q(0)/(1-x), where Q(k)= 1 + (x-1)*k!/(1 - x/(x + (x-1)*(k+1)!/Q(k+1))); (continued fraction).
%F (End)
%F E.g.f.: x/(1-x)*E(0)/2, where E(k)= 1 + 1/(1 - x/(x + (k+2)/E(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 01 2013
%F 1/(e - 1) = 1 - 1!/(1*3) - 2!/(3*10) - 3!/(10*41) - 4!/(41*206) - ... (see A056542 and A185108). - _Peter Bala_, Oct 09 2013
%F Conjecture: a(n) + (-n-1)*a(n-1) + (n-1)*a(n-2) = 0. - _R. J. Mathar_, Feb 16 2014
%F The e.g.f. f(x) = (exp(x)-1)/(1-x) satisfies the differential equation: (1-x)*f'(x) - (2-x)*f(x) + 1, from which we can obtain the recurrence:
%F a(n+1) = a(n) + n! + Sum_{k=1..n} (n!/k!)*a(k). The above conjectured recurrence can be obtained from the original recurrence or from the differential equation satisfied by f(x). - _Emanuele Munarini_, Jun 20 2014
%F Limit_{n -> infinity} a(n)/n! = exp(1) - 1. - _Carmine Suriano_, Jul 01 2015
%F Product_{n>=2} a(n)/(a(n)-1) = exp(1) - 1. See A091131. - _James R. Buddenhagen_, Jul 21 2019
%e [a(0), a(1), ...] = GAMMA(m+1,1)*exp(1) - GAMMA(m+1) = [exp(-1)*exp(1)-1, 2*exp(-1)*exp(1)-1, 5*exp(-1)*exp(1)-2, 16*exp(-1)*exp(1)-6, 65*exp(-1)*exp(1)-24, 326*exp(-1)*exp(1)-120, ...]. - _Stephen Crowley_, Jul 24 2009
%e From _Daniel Forgues_, Apr 25 2011: (Start)
%e n=0: {}: #{} = 0
%e n=1: {1}: #{()} = 1
%e n=2: {1,2}: #{(),(1),(2)} = 3
%e n=3: {1,2,3}: #{(),(1),(2),(3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)} = 10
%e (End)
%e x + 3*x^2 + 10*x^3 + 41*x^4 + 206*x^5 + 1237*x^6 + 8660*x^7 + 69281*x^8 + ...
%p A002627 := proc(n)
%p add( (n-j)!*binomial(n,j), j=1..n) ;
%p end proc:
%p seq(A002627(n),n=0..21) ; # _Zerinvary Lajos_, Jul 31 2006
%t FoldList[ #1*#2 + 1 &, 0, Range[21]] (* _Robert G. Wilson v_, Oct 11 2005 *)
%t RecurrenceTable[{a[0]==0,a[n]==n*a[n-1]+1},a,{n,30}] (* _Harvey P. Dale_, Mar 29 2015 *)
%o (PARI) a(n)= n!*sum(k=1,n, 1/k!); \\ _Joerg Arndt_, Apr 24 2011
%o (Haskell)
%o a002627 n = a002627_list !! n
%o a002627_list = 0 : map (+ 1) (zipWith (*) [1..] a002627_list)
%o -- _Reinhard Zumkeller_, Mar 24 2013
%o (Maxima) makelist(sum(n!/k!,k,1,n),n,0,40); /* _Emanuele Munarini_, Jun 20 2014 */
%o (Magma) I:=[1]; [0] cat [n le 1 select I[n] else n*Self(n-1)+1:n in [1..21]]; // _Marius A. Burtea_, Aug 07 2019
%Y Cf. A000522, A001113, A092582.
%Y Second diagonal of A059922, cf. A056542.
%Y Conjectured to give records in A130147.
%K nonn,easy,nice,changed
%O 0,3
%A _N. J. A. Sloane_
%E Comments from _Michael Somos_
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