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)
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
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.
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
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
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Sep 19 2000
STATUS
approved