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%I M0890 N0337 #136 Sep 08 2022 08:44:28
%S 2,3,8,30,144,840,5760,45360,403200,3991680,43545600,518918400,
%T 6706022400,93405312000,1394852659200,22230464256000,376610217984000,
%U 6758061133824000,128047474114560000,2554547108585472000,53523844179886080000,1175091669949317120000
%N a(n) = n! + (n-1)!.
%C Number of {12, 12*, 1*2, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
%C a(n) is the hook product of the shape (n, 1). - _Emeric Deutsch_, May 13 2004
%C From _Jaume Oliver Lafont_, Dec 01 2009: (Start)
%C (1+(x-1)*exp(x))/x = Sum_{k >= 1} x^k/a(k).
%C Setting x = 1 yields Sum_{k >= 1} 1/a(k) = 1. [Jolley eq 302] (End)
%C 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
%C 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
%C e is approximately a(n)/A000255(n-1) for large n. - _Dale Gerdemann_, Jul 26 2019
%C 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
%D L. B. W. Jolley, Summation of Series, Dover, 1961.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001048/b001048.txt">Table of n, a(n) for n=1..100</a>
%H Barry Balof and Helen Jenne, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Balof/balof22.html">Tilings, Continued Fractions, Derangements, Scramblings, and e</a>, Journal of Integer Sequences, Vol. 17 (2014), #14.2.7.
%H E. Biondi, L. Divieti, and G. Guardabassi, <a href="http://dx.doi.org/10.4153/CJM-1970-003-9">Counting paths, circuits, chains and cycles in graphs: A unified approach</a>, Canad. J. Math., Vol. 22, No. 1 (1970), pp. 22-35.
%H Richard K. Guy, <a href="/A002186/a002186.pdf">Letters to N. J. A. Sloane, June-August 1968</a>.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=97">Encyclopedia of Combinatorial Structures 97</a>.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=641">Encyclopedia of Combinatorial Structures 641</a>.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=101">Encyclopedia of Combinatorial Structures 101</a>.
%H Helen K. Jenne, <a href="https://www.whitman.edu/Documents/Academics/Mathematics/Jenne.pdf">Proofs you can count on</a>, Honors Thesis, Math. Dept., Whitman College, 2013.
%H B. D. Josephson and J. M. Boardman, <a href="https://archive.org/details/eureka-24/page/19/mode/2up">Problems Drive 1961</a>, Eureka, The Journal of the Archimedeans, Vol. 24 (1961), p. 20; <a href="https://www.archim.org.uk/eureka/archive/Eureka-24.pdf">entire volume</a>.
%H T. Mansour and J. West, <a href="http://arxiv.org/abs/math/0207204">Avoiding 2-letter signed patterns</a>, arXiv:math/0207204 [math.CO], 2002.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UniformSumDistribution.html">Uniform Sum Distribution</a>.
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F a(n) = (n+1)*(n-1)!.
%F E.g.f.: x/(1-x) - log(1-x). - _Ralf Stephan_, Apr 11 2004
%F 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
%F a(n) = (n+1)!/n. - Claude Lenormand (claude.lenormand(AT)free.fr), Aug 24 2003
%F Factorial expansion of 1: 1 = sum_{n > 0} 1/a(n) [Jolley eq 302]. - Claude Lenormand (claude.lenormand(AT)free.fr), Aug 24 2003
%F 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
%F a(n) = ((n+2)A052649(n) - A052649(n+1))/2. - _Gary Detlefs_, Dec 16 2009
%F 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
%F 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
%F a(n) = (n-1)*a(n-1) + (n-1)!. - _Bruno Berselli_, Feb 22 2017
%F 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
%F a(n) = 2*A000255(n-1) + A096654(n-2). - _Dale Gerdemann_, Jul 26 2019
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - 2/e (A334397). - _Amiram Eldar_, Jan 13 2021
%e 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
%p seq(n!+(n-1)!,n=1..25);
%t Table[n! + (n + 1)!, {n, 0, 30}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 14 2011 *)
%t Total/@Partition[Range[0, 20]!, 2, 1] (* _Harvey P. Dale_, Nov 29 2013 *)
%o (Magma) [Factorial(n)+Factorial(n+1): n in [0..30]]; // _Vincenzo Librandi_, Aug 08 2014
%o (PARI) a(n)=denominator(polcoeff((x-1)*exp(x+x*O(x^(n+1))),n+1)); \\ _Gerry Martens_, Aug 12 2015
%o (PARI) vector(30, n, (n+1)*(n-1)!) \\ _Michel Marcus_, Aug 12 2015
%Y Apart from initial terms, same as A059171.
%Y Equals the square root of the first right hand column of A162990. - _Johannes W. Meijer_, Jul 21 2009
%Y From a(2)=3 onward the second topmost row of arrays A276588 and A276955.
%Y Cf. sequences with formula (n + k)*n! listed in A282466, A334397.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, _R. K. Guy_
%E More terms from _James A. Sellers_, Sep 19 2000
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