A001048 a(n) = n! + (n-1)!. (Formerly M0890 N0337)

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

Offset

1,1

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)

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

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

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

Example

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

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 *)

Prog

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

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.

Sequence in context: A054104 A053556 A301737 * A141520 A072042 A160586

Adjacent sequences: A001045 A001046 A001047 * A001049 A001050 A001051

Keyword

nonn,easy

Author

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

Extensions

More terms from James A. Sellers, Sep 19 2000

Status

approved