A001048 a(n) = n! + (n-1)!.

2, 3, 8, 30, 144, 840, 5760, 45360, 403200, 3991680, 43545600, 518918400, 6706022400, 93405312000, 1394852659200, 22230464256000, 376610217984000, 6758061133824000, 128047474114560000, 2554547108585472000, 53523844179886080000, 1175091669949317120000

Offset: 1

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• easy

Comments

Number of {12, 12*, 1*2, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.

a(n) is the hook product of the shape (n, 1). - Emeric Deutsch, May 13 2004

From Jaume Oliver Lafont, Dec 01 2009: (Start)

(1+(x-1)*exp(x))/x = Sum_{k >= 1} x^k/a(k).

Setting x = 1 yields Sum_{k >= 1} 1/a(k) = 1. [Jolley eq 302] (End)

With regard to the comment by Jaume Oliver Lafont: P(n) = 1/a(n) is a probability distribution, with all values given as unit fractions. This distribution is connected to the Irwin-Hall distribution: Consider successively drawn random numbers, uniformly distributed in [0,1]. 1/a(n) is the probability for the sum of the random numbers exceeding 1 exactly with the (n+1)-th summand. P(n) has mean e-1 and variance 3e-e^2. From this we get e as the expected number of summands. - Manfred Boergens, May 20 2024

For n >= 2, a(n) is the size of the largest conjugacy class of the symmetric group on n + 1 letters. Equivalently, the maximum entry in each row of A036039. - Geoffrey Critzer, May 19 2013

In factorial base representation (A007623) the terms are written as: 10, 11, 110, 1100, 11000, 110000, ... From a(2) = 3 = "11" onward each term begins always with two 1's, followed by n-2 zeros. - Antti Karttunen, Sep 24 2016

e is approximately a(n)/A000255(n-1) for large n. - Dale Gerdemann, Jul 26 2019

a(n) is the number of permutations of [n+1] in which all the elements of [n] are cycle-mates, that is, 1,..,n are all in the same cycle. This result is readily shown after noting that the elements of [n] can be members of a n-cycle or an (n+1)-cycle. Hence a(n)=(n-1)!+n!. See an example below. - Dennis P. Walsh, May 24 2020

Examples

			For n=3, a(3) counts the 8 permutations of [4] with 1,2, and 3 all in the same cycle, namely, (1 2 3)(4), (1 3 2)(4), (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 2 4 3), (1 4 2 3), and (1 4 3 2). - _Dennis P. Walsh_, May 24 2020
		

References

• L. B. W. Jolley, Summation of Series, Dover, 1961.
• 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

• T. D. Noe, Table of n, a(n) for n=1..100
• Barry Balof and Helen Jenne, Tilings, Continued Fractions, Derangements, Scramblings, and e, Journal of Integer Sequences, Vol. 17 (2014), #14.2.7.
• E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math., Vol. 22, No. 1 (1970), pp. 22-35.
• Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 97.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 641.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 101.
• Helen K. Jenne, Proofs you can count on, Honors Thesis, Math. Dept., Whitman College, 2013.
• B. D. Josephson and J. M. Boardman, Problems Drive 1961, Eureka, The Journal of the Archimedeans, Vol. 24 (1961), p. 20; entire volume.
• T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
• Eric Weisstein's World of Mathematics, Uniform Sum Distribution.
• Index entries for sequences related to factorial base representation
• Index entries for sequences related to factorial numbers

Crossrefs

Apart from initial terms, same as A059171.

Equals the square root of the first right hand column of A162990. - Johannes W. Meijer, Jul 21 2009

From a(2)=3 onward the second topmost row of arrays A276588 and A276955.

Cf. sequences with formula (n + k)*n! listed in A282466, A334397.

Programs

Magma

[Factorial(n)+Factorial(n+1): n in [0..30]]; // Vincenzo Librandi, Aug 08 2014

Maple

seq(n!+(n-1)!,n=1..25);

Mathematica

Table[n! + (n + 1)!, {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Apr 14 2011 *)
Total/@Partition[Range[0, 20]!, 2, 1] (* Harvey P. Dale, Nov 29 2013 *)

PARI

a(n)=denominator(polcoeff((x-1)*exp(x+x*O(x^(n+1))),n+1)); \\ Gerry Martens, Aug 12 2015

PARI

vector(30, n, (n+1)*(n-1)!) \\ Michel Marcus, Aug 12 2015

Formula

a(n) = (n+1)*(n-1)!.

E.g.f.: x/(1-x) - log(1-x). - Ralf Stephan, Apr 11 2004

The sequence 1, 3, 8, ... has g.f. (1+x-x^2)/(1-x)^2 and a(n) = n!(n + 2 - 0^n) = n!A065475(n) (offset 0). - Paul Barry, May 14 2004

a(n) = (n+1)!/n. - Claude Lenormand (claude.lenormand(AT)free.fr), Aug 24 2003

Factorial expansion of 1: 1 = sum_{n > 0} 1/a(n) [Jolley eq 302]. - Claude Lenormand (claude.lenormand(AT)free.fr), Aug 24 2003

a(1) = 2, a(2) = 3, D-finite recurrence a(n) = (n^2 - n - 2)*a(n-2) for n >= 3. - Jaume Oliver Lafont, Dec 01 2009

a(n) = ((n+2)A052649(n) - A052649(n+1))/2. - Gary Detlefs, Dec 16 2009

G.f.: U(0) where U(k) = 1 + (k+1)/(1 - x/(x + 1/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 25 2012

G.f.: 2*(1+x)/x/G(0) - 1/x, where G(k)= 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013

a(n) = (n-1)*a(n-1) + (n-1)!. - Bruno Berselli, Feb 22 2017

a(1)=2, a(2)=3, D-finite recurrence a(n) = (n-1)*a(n-1) + (n-2)*a(n-2). - Dale Gerdemann, Jul 26 2019

a(n) = 2*A000255(n-1) + A096654(n-2). - Dale Gerdemann, Jul 26 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - 2/e (A334397). - Amiram Eldar, Jan 13 2021

Extensions

More terms from James Sellers, Sep 19 2000