A002627 a(n) = n*a(n-1) + 1, a(0) = 0.

0, 1, 3, 10, 41, 206, 1237, 8660, 69281, 623530, 6235301, 68588312, 823059745, 10699776686, 149796873605, 2246953104076, 35951249665217, 611171244308690, 11001082397556421, 209020565553572000, 4180411311071440001, 87788637532500240022

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

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

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

Sums of rows of the triangle in A173333, n > 0. - Reinhard Zumkeller, Feb 19 2010

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

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

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

a(n) = q(n,1) for n >= 1, where the polynomials q are defined at A248669. - Clark Kimberling, Oct 11 2014

a(n) is the number of quasilinear weak orderings on {1,...,n}. - J. Devillet, Dec 22 2017

Examples

			[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
From _Daniel Forgues_, Apr 25 2011: (Start)
  n=0: {}: #{} = 0
  n=1: {1}: #{()} = 1
  n=2: {1,2}: #{(),(1),(2)} = 3
  n=3: {1,2,3}: #{(),(1),(2),(3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)} = 10
(End)
x + 3*x^2 + 10*x^3 + 41*x^4 + 206*x^5 + 1237*x^6 + 8660*x^7 + 69281*x^8 + ...
		

References

• D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.
• 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

• Seiichi Manyama, Table of n, a(n) for n = 0..449 (terms 0..100 from T. D. Noe)
• Sanka Balasuriya, Igor E. Shparlinski and Arne Winterhof, An average bound for character sums with some counter-dependent recurrence sequences, Rocky Mt. J. Math. 39, No. 5, 1403-1409 (2009).
• Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
• Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
• Jimmy Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 150
• Nicholas Kapoor and P. Christopher Staecker, Ahead of the Count: An Algorithm for Probabilistic Prediction of Instant Runoff (IRV) Elections, arXiv:2405.09009 [cs.CY], 2024. See p. 11.
• Daljit Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70. [Annotated scanned copy]
• Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 5.

Crossrefs

Cf. A000522, A001113, A092582.

Second diagonal of A059922, cf. A056542.

Conjectured to give records in A130147.

Programs

Haskell

a002627 n = a002627_list !! n
a002627_list = 0 : map (+ 1) (zipWith (*) [1..] a002627_list)
-- Reinhard Zumkeller, Mar 24 2013

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

Maple

A002627 := proc(n)
    add( (n-j)!*binomial(n,j), j=1..n) ;
end proc:
seq(A002627(n),n=0..21) ; # Zerinvary Lajos, Jul 31 2006

Mathematica

FoldList[ #1*#2 + 1 &, 0, Range[21]] (* Robert G. Wilson v, Oct 11 2005 *)
RecurrenceTable[{a[0]==0,a[n]==n*a[n-1]+1},a,{n,30}] (* Harvey P. Dale, Mar 29 2015 *)

Maxima

makelist(sum(n!/k!,k,1,n),n,0,40); /* Emanuele Munarini, Jun 20 2014 */

PARI

a(n)= n!*sum(k=1,n, 1/k!); \\ Joerg Arndt, Apr 24 2011

Formula

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

a(n) = A000522(n) - n!. - Michael Somos, Mar 26 1999

a(n) = floor( n! * (e-1) ), n >= 1. - Amarnath Murthy, Mar 08 2002

E.g.f.: (exp(x)-1)/(1-x). - Mario Catalani (mario.catalani(AT)unito.it), Feb 06 2003

Binomial transform of A002467. - Ross La Haye, Sep 21 2004

a(n) = Sum_{j=1..n} (n-j)!*binomial(n,j). - Zerinvary Lajos, Jul 31 2006

a(n) = 1 + Sum_{k=0..n-1} k*a(k). - Benoit Cloitre, Jul 26 2008

a(m) = Integral_{s=0..oo} ((1+s)^m - s^m)*exp(-s) = GAMMA(m+1,1) * exp(1) - GAMMA(m+1). - Stephen Crowley, Jul 24 2009

From Sergei N. Gladkovskii, Jul 05 2012: (Start)

a(n+1) = A000522(n) + A001339(n) - A000142(n+1);

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). (End)

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

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

Conjecture: a(n) + (-n-1)*a(n-1) + (n-1)*a(n-2) = 0. - R. J. Mathar, Feb 16 2014

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:

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

Limit_{n -> oo} a(n)/n! = exp(1) - 1. - Carmine Suriano, Jul 01 2015

Product_{n>=2} a(n)/(a(n)-1) = exp(1) - 1. See A091131. - James R. Buddenhagen, Jul 21 2019

a(n) = Sum_{k=0..n-1} k!*binomial(n,k). - Ridouane Oudra, Jun 17 2025

Extensions

Comments from Michael Somos