A334397 -id:A334397 - cp's OEIS frontend

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

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

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

			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

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).

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

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.

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

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

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 *)

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

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

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

More terms from James Sellers, Sep 19 2000

A309419 Decimal expansion of e/(e-2).

Original entry on oeis.org

3, 7, 8, 4, 4, 2, 2, 3, 8, 2, 3, 5, 4, 6, 6, 5, 6, 2, 8, 7, 5, 3, 1, 0, 5, 7, 5, 6, 9, 5, 9, 6, 3, 3, 0, 5, 6, 7, 4, 7, 9, 5, 6, 7, 7, 0, 6, 3, 0, 5, 7, 4, 2, 4, 7, 1, 8, 2, 6, 4, 9, 1, 3, 4, 1, 6, 6, 5, 5, 9, 1, 4, 0, 9, 2, 3, 2, 2, 1, 8, 5, 3, 3, 8, 3, 4, 2, 1, 1, 7, 4, 5, 3, 5, 2, 2, 5, 9, 9, 7, 7, 7, 7, 1, 3, 7

Offset: 1

Alois P. Heinz, Jul 30 2019

This can be computed using a recursion formula discovered by an algorithm called "The Ramanujan Machine":
                             1
    e/(e-2)  =  4 - --------------------
                               2
                    5 - ----------------
                                 3
                        6 - ------------
                                   4
                            7 - --------
                                 8 - ...  .
  For a proof by humans see the arXiv:1907.00205 preprint linked below.

			3.78442238235466562875310575695963305674795677063...

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Gal Raayoni, George Pisha, Yahel Manor, Uri Mendlovic, Doron Haviv, Yaron Hadad, and Ido Kaminer, The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants, arXiv:1907.00205 [cs.LG], 2019-2020.
The Ramanujan Machine, Using algorithms to discover new mathematics.

Cf. A001113, A309091, A309420, A334397.

nn:= 126: # number of digits
b:= i-> `if`(i

RealDigits[E/(E-2), 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

Equals 1/A334397.

cp's OEIS Frontend

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

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A309419 Decimal expansion of e/(e-2).

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A371934 Decimal expansion of Sum_{k>=0} (-1)^k / ((k+1)(2k)!).

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cp's OEIS Frontend

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

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A309419 Decimal expansion of e/(e-2).

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A371934 Decimal expansion of Sum_{k>=0} (-1)^k / ((k+1)*(2*k)!).

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A371934 Decimal expansion of Sum_{k>=0} (-1)^k / ((k+1)(2k)!).