Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I M1221 N0469 #549 Oct 27 2024 11:38:15
%S 1,1,2,4,10,26,76,232,764,2620,9496,35696,140152,568504,2390480,
%T 10349536,46206736,211799312,997313824,4809701440,23758664096,
%U 119952692896,618884638912,3257843882624,17492190577600,95680443760576,532985208200576,3020676745975552
%N Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.
%C a(n) is also the number of n X n symmetric permutation matrices.
%C a(n) is also the number of matchings (Hosoya index) in the complete graph K(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
%C a(n) is also the number of independent vertex sets and vertex covers in the n-triangular graph. - _Eric W. Weisstein_, May 22 2017
%C Equivalently, this is the number of graphs on n labeled nodes with degrees at most 1. - _Don Knuth_, Mar 31 2008
%C a(n) is also the sum of the degrees of the irreducible representations of the symmetric group S_n. - Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001
%C a(n) is the number of partitions of a set of n distinguishable elements into sets of size 1 and 2. - _Karol A. Penson_, Apr 22 2003
%C Number of tableaux on the edges of the star graph of order n, S_n (sometimes T_n). - _Roberto E. Martinez II_, Jan 09 2002
%C The Hankel transform of this sequence is A000178 (superfactorials). Sequence is also binomial transform of the sequence 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, ... (A001147 with interpolated zeros). - _Philippe Deléham_, Jun 10 2005
%C Row sums of the exponential Riordan array (e^(x^2/2),x). - _Paul Barry_, Jan 12 2006
%C a(n) is the number of nonnegative lattice paths of upsteps U = (1,1) and downsteps D = (1,-1) that start at the origin and end on the vertical line x = n in which each downstep (if any) is marked with an integer between 1 and the height of its initial vertex above the x-axis. For example, with the required integer immediately preceding each downstep, a(3) = 4 counts UUU, UU1D, UU2D, U1DU. - _David Callan_, Mar 07 2006
%C Equals row sums of triangle A152736 starting with offset 1. - _Gary W. Adamson_, Dec 12 2008
%C Proof of the recurrence relation a(n) = a(n-1) + (n-1)*a(n-2): number of involutions of [n] containing n as a fixed point is a(n-1); number of involutions of [n] containing n in some cycle (j, n), where 1 <= j <= n-1, is (n-1) times the number of involutions of [n] containing the cycle (n-1 n) = (n-1)*a(n-2). - _Emeric Deutsch_, Jun 08 2009
%C Number of ballot sequences (or lattice permutations) of length n. A ballot sequence B is a string such that, for all prefixes P of B, h(i) >= h(j) for i < j, where h(x) is the number of times x appears in P. For example, the ballot sequences of length 4 are 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1231, and 1234. The string 1221 does not appear in the list because in the 3-prefix 122 there are two 2's but only one 1. (Cf. p. 176 of Bruce E. Sagan: "The Symmetric Group"). - _Joerg Arndt_, Jun 28 2009
%C Number of standard Young tableaux of size n; the ballot sequences are obtained as a length-n vector v where v_k is the (number of the) row in which the number r occurs in the tableaux. - _Joerg Arndt_, Jul 29 2012
%C Number of factorial numbers of length n-1 with no adjacent nonzero digits. For example the 10 such numbers (in rising factorial radix) of length 3 are 000, 001, 002, 003, 010, 020, 100, 101, 102, and 103. - _Joerg Arndt_, Nov 11 2012
%C Also called restricted Stirling numbers of the second kind (see Mezo). - _N. J. A. Sloane_, Nov 27 2013
%C a(n) is the number of permutations of [n] that avoid the consecutive patterns 123 and 132. Proof. Write a self-inverse permutation in standard cycle form: smallest entry in each cycle in first position, first entries decreasing. For example, (6,7)(3,4)(2)(1,5) is in standard cycle form. Then erase parentheses. This is a bijection to the permutations that avoid consecutive 123 and 132 patterns. - _David Callan_, Aug 27 2014
%C Getu (1991) says these numbers are also known as "telephone numbers". - _N. J. A. Sloane_, Nov 23 2015
%C a(n) is the number of elements x in S_n such that x^2 = e where e is the identity. - _Jianing Song_, Aug 22 2018
%C a(n) is the number of congruence orbits of upper-triangular n X n matrices on skew-symmetric matrices, or the number of Borel orbits in largest sect of the type DIII symmetric space SO_{2n}(C)/GL_n(C). Involutions can also be thought of as fixed-point-free partial involutions. See [Bingham and Ugurlu] link. - _Aram Bingham_, Feb 08 2020
%C From _Thomas Anton_, Apr 20 2020: (Start)
%C Apparently a(n) = b*c where b is odd iff a(n+b) (when a(n) is defined) is divisible by b.
%C Apparently a(n) = 2^(f(n mod 4)+floor(n/4))*q where f:{0,1,2,3}->{0,1,2} is given by f(0),f(1)=0, f(2)=1 and f(3)=2 and q is odd. (End)
%C From _Iosif Pinelis_, Mar 12 2021: (Start)
%C a(n) is the n-th initial moment of the normal distribution with mean 1 and variance 1. This follows because the moment generating function of that distribution is the e.g.f. of the sequence of the a(n)'s.
%C The recurrence a(n) = a(n-1) + (n-1)*a(n-2) also follows, by writing E(Z+1)^n=EZ(Z+1)^(n-1)+E(Z+1)^(n-1), where Z is a standard normal random variable, and then taking the first of the latter two integrals by parts. (End)
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 32, 911.
%D S. Chowla, The asymptotic behavior of solutions of difference equations, in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), Vol. I, 377, Amer. Math. Soc., Providence, RI, 1952.
%D W. Fulton, Young Tableaux, Cambridge, 1997.
%D D. E. Knuth, The Art of Computer Programming, Vol. 3, Section 5.1.4, p. 65.
%D L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
%D T. Muir, A Treatise on the Theory of Determinants. Dover, NY, 1960, p. 6.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 86.
%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).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
%H Alois P. Heinz, <a href="/A000085/b000085.txt">Table of n, a(n) for n = 0..800</a> (first 101 terms from T. D. Noe)
%H Ahmad Zainy Al-Yasry, <a href="https://arxiv.org/abs/1901.03640">Short Steps in Noncommutative Geometry</a>, arXiv:1901.03640 [math.OA], 2019.
%H T. Amdeberhan and V. H. Moll, <a href="http://129.81.170.14/~vhm/papers_html/invo-may12.pdf">Involutions and their progenies</a>, 2014.
%H David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a> arXiv:0907.0513 [math.CO], 2009.
%H Joerg Arndt, <a href="http://jjj.de/pub/arndt-rand-perm-thesis.pdf">Generating Random Permutations</a>, PhD thesis, Australian National University, Canberra, Australia, (2010).
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp. 279-280.
%H C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, <a href="http://algo.inria.fr/banderier/Papers/DiscMath99.ps">Generating Functions for Generating Trees</a>, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
%H D. Barsky, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05barsky.html">Analyse p-adique et suites classiques de nombres</a>, Sem. Loth. Comb. B05b (1981) 1-21.
%H C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, <a href="https://arxiv.org/abs/math/0506334">On the X-rays of permutations</a>, arXiv:math/0506334 [math.CO], 2005.
%H E. A. Bender and S. G. Williamson, <a href="http://www.math.ucsd.edu/~ebender/CombText/index.html">Foundations of Combinatorics with Applications</a> (see Chap. 2, Example 2.9, pp. 47-48, including Theorem 2.2, a derived formula for a(n)). [From _Rick L. Shepherd_, Sep 02 2009]
%H S. Bilotta, E. Pergola, R. Pinzani, and S. Rinaldi, <a href="http://arxiv.org/abs/1301.2967">Recurrence relations versus succession rules</a>, arXiv preprint arXiv:1301.2967 [cs.DM], 2013. - From _N. J. A. Sloane_, Feb 12 2013
%H S. Bilotta, E. Pergola, R. Pinzani, and S. Rinaldi, <a href="http://dx.doi.org/10.1007/978-3-319-15579-1_39">Recurrence Relations, Succession Rules, and the Positivity Problem</a>, in Language and Automata Theory and Applications, 9th International Conference, LATA 2015, Nice, France, March 2-6, 2015, Proceedings, Pages 499-510, Lecture Notes Comp. Sci. Vol. 8977.
%H Aram Bingham and Özlem Uğurlu, <a href="https://arxiv.org/abs/1907.08875">Sects, rooks, pyramids, partitions and paths for type DIII clans</a>, arXiv:1907.08875 [math.CO], 2019.
%H Anders Björner and Richard P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.pdf">A combinatorial miscellany</a>, 2010.
%H P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, <a href="http://arXiv.org/abs/quant-ph/0405103">Combinatorial field theories via boson normal ordering</a>, arXiv:quant-ph/0405103, 2004-2006.
%H P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, <a href="http://dx.doi.org/10.1063/1.1904161">Some useful combinatorial formulas for bosonic operators</a>, J. Math. Phys. 46, 052110 (2005) (6 pages).
%H Tobias Boege and Thomas Kahle, <a href="https://arxiv.org/abs/1902.11260">Construction Methods for Gaussoids</a>, arXiv:1902.11260 [math.CO], 2019.
%H D. Bundala, M. Codish, L. Cruz-Filipe et al., <a href="http://arxiv.org/abs/1412.5302">Optimal-Depth Sorting Networks</a>, arXiv preprint arXiv:1412.5302 [cs.DS], 2014.
%H Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H D. Castellanos, <a href="http://www.fq.math.ca/Scanned/27-5/castellanos.pdf">A generalization of Binet's formula and some of its consequences</a>, Fib. Quart., 27 (1989), 424-438.
%H C.-O. Chow, <a href="http://dx.doi.org/10.1016/j.disc.2006.04.018">Counting involutory, unimodal and alternating signed permutations</a>, Discr. Math. 306 (2006), 2222-2228.
%H S. Chowla, and I.N. Hernstein, W.K. Moore, <a href="http://dx.doi.org/10.4153/CJM-1951-038-3">On recursions connected with symmetric groups. I</a>, Canadian Journal of Mathematics, 3 (1951), 328-334.
%H M. Codish, L. Cruz-Filipe and P. Schneider-Kamp, <a href="http://arxiv.org/abs/1404.0948">The Quest for Optimal Sorting Networks: Efficient Generation of Two-Layer Prefixes</a>, arXiv preprint arXiv:1404.0948 [cs.DS], 2014.
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/">Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras</a>
%H Filippo Disanto and Thomas Wiehe, <a href="http://arxiv.org/abs/1210.6908">Some instances of a sub-permutation problem on pattern avoiding permutations</a>, arXiv preprint arXiv:1210.6908 [math.CO], 2012.
%H I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., <a href="http://arxiv.org/abs/1408.2021">Enumeration of idempotents in diagram semigroups and algebras</a>, arXiv preprint arXiv:1408.2021 [math.GR], 2014.
%H I. Dolinka, J. East, and R. D. Gray, <a href="http://arxiv.org/abs/1512.02279">Motzkin monoids and partial Brauer monoids</a>, arXiv preprint arXiv:1512.02279 [math.GR], 2015-2016.
%H R. Donaghey, <a href="http://dx.doi.org/10.1016/0097-3165(76)90060-1">Binomial self-inverse sequences and tangent coefficients</a>, J. Combin. Theory, Series A, 21 (1976), 155-163.
%H S. Dulucq and J.-G. Penaud, <a href="http://dx.doi.org/10.1016/0012-365X(93)90326-O">Cordes, arbres et permutations</a>, Discrete Math. 117 (1993), no. 1-3, 89-105.
%H Shalosh B. Ekhad, Manuel Kauers, and Doron Zeilberger, <a href="https://arxiv.org/abs/2410.16334">The O(1/n^85) Asymptotic expansion of OEIS sequence A85</a>, arxiv preprint arXiv:2410.16334 [math.CO], October 2024.
%H Shalosh B. Ekhad and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/ssyt.html">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>
%H Shalosh B. Ekhad and Doron Zeilberger, <a href="/A000085/a000085_1.pdf">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, [Local copy, pdf file only, no active links]
%H Shalosh B. Ekhad and Doron Zeilberger, <a href="https://arxiv.org/abs/1202.6229">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, arXiv preprint arXiv:1202.6229 [math.CO], 2012. [This is a different document from the one with the same title on Doron Zeilberger's web site]
%H Shalosh B. Ekhad, Manuel Kauers, and Doron Zeilberger, <a href="https://arxiv.org/abs/2410.16334">The O(1/n^85) Asymptotic expansion of OEIS sequence A85</a>, arXiv:2410.16334 [math.CO], 2024.
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StandardTableaux">Standard Young tableaux</a>
%H Amy M. Fu and Frank Z. K. Li, <a href="https://arxiv.org/abs/1809.07465">Joint Distributions of Permutation Statistics and the Parabolic Cylinder Functions</a>, arXiv:1809.07465 [math.CO], 2018.
%H Mohammad Ganjtabesh, Armin Morabbi and Jean-Marc Steyaert, <a href="http://www.lifl.fr/SEQUOIA/Arena/Presentations/Ganjtabesh.pdf">Enumerating the number of RNA structures</a>
%H S. Garrabrant and I. Pak, <a href="http://arxiv.org/abs/1407.8222">Counting with irrational tiles</a>, arXiv:1407.8222 [math.CO], 2014.
%H S. Getu, <a href="http://www.jstor.org/stable/2690455">Evaluating determinants via generating functions</a>, Mathematics Magazine 64 (1991), 45-53.
%H J. Gilder, <a href="http://www.jstor.org/stable/3615831">Rooks inviolate revisited II</a>, Math. Gaz., 70 (1986), 44-45.
%H A. M. Goyt, <a href="https://arxiv.org/abs/math/0603481">Avoidance of partitions of a 3-element set</a>, arXiv:math/0603481 [math.CO], 2006.
%H Greenhill, Catherine; McKay, Brendan D., <a href="https://doi.org/10.1016/j.laa.2011.03.052">Counting loopy graphs with given degree</a>, Linear Algebra Appl. 436, No. 4, 901-926 (2012)
%H H. Gupta, <a href="http://dx.doi.org/10.1215/S0012-7094-68-03567-9">Enumeration of symmetric matrices</a>, Duke Math. J., 35 (1968), vol 3, 653-659
%H H. Gupta, <a href="/A000085/a000085.pdf">Enumeration of symmetric matrices</a> (annotated scanned copy)
%H C. Gutschwager, <a href="https://dx.doi.org/10.1016/j.ejc.2010.05.008">Reduced Kronecker products which are multiplicity free or contain only a few components</a>, Eur. J. Combinat. 31 (2010) 1996-2005 doi:10.1016/j.ejc.2010.05.008, Remark 2.13.
%H T. Halverson and M. Reeks, <a href="http://arxiv.org/abs/1302.6150">Gelfand Models for Diagram Algebras</a>, arXiv preprint arXiv:1302.6150 [math.RT], 2013.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0409152">A product formula and combinatorial field theory</a>, arXiv:quant-ph/0409152, 2004.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=17">Encyclopedia of Combinatorial Structures 17</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=22">Encyclopedia of Combinatorial Structures 22</a>
%H András Kaszanyitzky, <a href="https://arxiv.org/abs/1805.11532">The GraftalLace Cellular Automaton</a>, arXiv:1805.11532 [nlin.CG], 2018.
%H Manuel Kauers, Doron Zeilberger, <a href="https://arxiv.org/abs/2006.10205">Counting Standard Young Tableaux With Restricted Runs</a>, arXiv:2006.10205 [math.CO], 2020.
%H Dongsu Kim and Jang Soo Kim, <a href="http://dx.doi.org/10.1016/j.jcta.2009.08.002">A combinatorial approach to the power of 2 in the number of involutions</a>, J. Comb. Theory Ser. A 117 (8) (2010): 1082-1094
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 227.
%H P. L. Krapivsky and J. M. Luck, <a href="https://arxiv.org/abs/1902.04365">Coverage fluctuations in theater models</a>, arXiv:1902.04365 [cond-mat.stat-mech], 2019.
%H M. B. Kutler and C. R. Vinroot, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Vinroot/vinroot3.html">On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups</a>, JIS 13 (2010) #10.3.6.
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H L. C. Larson, <a href="/A000900/a000900_1.pdf">The number of essentially different nonattacking rook arrangements</a>, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
%H J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H Steven Linton, James Propp, Tom Roby, and Julian West, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Roby/roby4.html">Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.
%H E. Lucas, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k29021h">Théorie des Nombres</a>, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
%H E. Lucas, <a href="/A000899/a000899.pdf">Théorie des nombres</a> (annotated scans of a few selected pages)
%H T. Mansour and M. Shattuck, <a href="http://dx.doi.org/10.2298/AADM121130023M">Partial matchings and pattern avoidance</a>, Appl. Anal. Discrete Math. (to appear, 2012); - From _N. J. A. Sloane_, Jan 04 2013
%H I. Mezo, <a href="http://arxiv.org/abs/1308.1637">Periodicity of the last digits of some combinatorial sequences</a>, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mezo/mezo19.html">J. Int. Seq. 17 (2014) #14.1.1</a>.
%H F. L. Miksa, L. Moser and M. Wyman, <a href="http://dx.doi.org/10.4153/CMB-1958-010-2">Restricted partitions of finite sets</a>, Canad. Math. Bull., 1 (1958), 87-96.
%H L. Moser and M. Wyman, <a href="http://cms.math.ca/10.4153/CJM-1955-021-8">On solutions of x^d = 1 in symmetric groups</a>, Canad. J. Math., 7 (1955), 159-168.
%H T. Mueller, <a href="http://dx.doi.org/10.1007/BF01195003">Finite group actions and asymptotic expansion of e^P(z)</a>, Combinatorica, 17 (4) (1997), 523-554.
%H Igor Pak, <a href="https://arxiv.org/abs/1803.06636">Complexity problems in enumerative combinatorics</a>, arXiv:1803.06636 [math.CO], 2018.
%H I. Pak, G. Panova, and E. Vallejo, <a href="http://arxiv.org/abs/1304.0738">Kronecker products, characters, partitions, and the tensor square conjectures</a>, arXiv:1304.0738 [math.CO], 2013.
%H D. Panyushev, <a href="http://arxiv.org/abs/1407.6857">On the orbits of a Borel subgroup in abelian ideals</a>, arXiv preprint arXiv:1407.6857 [math.AG], 2014.
%H P. Peart and W.-J. Woan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/PEART/peart1.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H R. Petuchovas, <a href="https://arxiv.org/abs/1611.02934">Asymptotic analysis of the cyclic structure of permutations</a>, arXiv:1611.02934 [math.CO], p. 6, 2016.
%H R. A. Proctor, <a href="https://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way for Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.
%H J. L. Ramírez and G. N. Rubiano, <a href="http://www.mathematica-journal.com/2014/02/properties-and-generalizations-of-the-fibonacci-word-fractal/">Properties and Generalizations of the Fibonacci Word Fractal</a>, The Mathematica Journal, Vol. 16 (2014).
%H J. Riordan, <a href="/A001813/a001813.pdf">Letter to N. J. A. Sloane, Feb 03 1975</a> (with notes by njas)
%H R. W. Robinson, <a href="http://dx.doi.org/10.1007/BFb0097382">Counting arrangements of bishops</a>, Lect. Notes Math. 560 (1976), 198-214. See D_n.
%H H. A. Rothe, <a href="http://www.e-rara.ch/zut/content/pageview/1341041">Ueber Permutationen, in Beziehung auf die Stellen ihrer Elemente. Anwendung der daraus abgeleiteten Sätze auf das Eliminationsproblem</a>, in Carl Hindenberg, ed., Sammlung Combinatorisch-Analytischer Abhandlungen, pp. 263-305, Bey G. Fleischer dem jüngern, 1800; see p. 282.
%H Ahmad Sabri and Vincent Vajnovszki, <a href="http://ceur-ws.org/Vol-2113/paper21.pdf">Exhaustive generation for ballot sequences in lexicographic and Gray code order</a>, CEUR Workshop Proceedings, 2018.
%H Ahmad Sabri and Vincent Vajnovszki, <a href="https://doi.org/10.1515/puma-2015-0035">On the exhaustive generation of generalized ballot sequences in lexicographic and Gray code order</a>, Pure Mathematics and Applications (2019) Vol. 28, Issue 1, 109-119.
%H A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, <a href="http://arXiv.org/abs/quant-ph/0310174">Combinatorial physics, normal order and model Feynman graphs</a>, arXiv:quant-ph/0310174, 2003.
%H Bridget Eileen Tenner, <a href="https://arxiv.org/abs/2004.03286">Star factorizations and noncrossing partitions</a>, arXiv:2004.03286 [math.CO], 2020.
%H Robert F. Tichy and Stephan Wagner, <a href="http://dx.doi.org/10.1089/cmb.2005.12.1004">Extremal Problems for Topological Indices in Combinatorial Chemistry</a>, Journal of Computational Biology. September 2005, 12(7): 1004-1013.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HosoyaIndex.html">Hosoya Index</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PermutationInvolution.html">Permutation Involution</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/YoungTableau.html">Young Tableau</a>
%H Allan Wentink, <a href="/A000085/a000085_2.pdf">Symmetry with Independent Blocks</a>, on the late Ralph E. Griswold webpage of sample-chapters of an unfinished book about weaving, http://www.cs.arizona.edu/patterns/weaving/webdocs.html. [Cached copy]
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Telephone_number_(mathematics)">Telephone numbers</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Young_tableaux">Young tableau</a>.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F D-finite with recurrence a(0) = a(1) = 1, a(n) = a(n-1) + (n-1)*a(n-2) for n>1.
%F E.g.f.: exp(x+x^2/2).
%F a(n) = a(n-1) + A013989(n-2) = A013989(n)/(n+1) = 1+A001189(n).
%F a(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)!*2^k*k!).
%F a(m+n) = Sum_{k>=0} k!*binomial(m, k)*binomial(n, k)*a(m-k)*a(n-k). - _Philippe Deléham_, Mar 05 2004
%F For n>1, a(n) = 2*(A000900(n) + A000902(floor(n/2))). - _Max Alekseyev_, Oct 31 2015
%F The e.g.f. y(x) satisfies y^2 = y''y' - (y')^2.
%F a(n) ~ c*(n/e)^(n/2)exp(n^(1/2)) where c=2^(-1/2)exp(-1/4). [Chowla]
%F a(n) = HermiteH(n, 1/(sqrt(2)*i))/(-sqrt(2)*i)^n, where HermiteH are the Hermite polynomials. - _Karol A. Penson_, May 16 2002
%F a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2. - _Paul Barry_, Jan 12 2006
%F For asymptotics see the Robinson paper.
%F a(n) = Sum_{m=0..n} A099174(n,m). - _Roger L. Bagula_, Oct 06 2006
%F O.g.f.: A(x) = 1/(1-x-1*x^2/(1-x-2*x^2/(1-x-3*x^2/(1-... -x-n*x^2/(1- ...))))) (continued fraction). - _Paul D. Hanna_, Jan 17 2006
%F From _Gary W. Adamson_, Dec 29 2008: (Start)
%F a(n) = (n-1)*a(n-2) + a(n-1); e.g., a(7) = 232 = 6*26 + 76.
%F Starting with offset 1 = eigensequence of triangle A128229. (End)
%F a(n) = (1/sqrt(2*Pi))*Integral_{x=-oo..oo} exp(-x^2/2)*(x+1)^n. - _Groux Roland_, Mar 14 2011
%F Row sums of |A096713|. a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+2*x)*d/dx. Cf. A047974 and A080599. - _Peter Bala_, Dec 07 2011
%F From _Sergei N. Gladkovskii_, Dec 03 2011 - Oct 28 2013: (Start)
%F Continued fractions:
%F E.g.f.: 1+x*(2+x)/(2*G(0)-x*(2+x)) where G(k)=1+x*(x+2)/(2+2*(k+1)/G(k+1)).
%F G.f.: 1/(U(0) - x) where U(k) = 1 + x*(k+1) - x*(k+1)/(1 - x/U(k+1)).
%F G.f.: 1/Q(0) where Q(k) = 1 + x*k - x/(1 - x*(k+1)/Q(k+1)).
%F G.f.: -1/(x*Q(0)) where Q(k) = 1 - 1/x - (k+1)/Q(k+1).
%F G.f.: T(0)/(1-x) where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x)^2/T(k+1)). (End)
%F a(n) ~ sqrt(2)/2 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - _Vaclav Kotesovec_, Mar 07 2014
%F a(n) = Sum_{k=0..n} s(n,k)*(-1)^(n-k)*2^k*B(k,1/2), where the s(n,k) are (signless) Stirling numbers of the first kind, and the B(n,x) = Sum_{k=0..n} S(n,k)*x^k are the Stirling polynomials, where the S(n,k) are the Stirling numbers of the second kind. - _Emanuele Munarini_, May 16 2014
%F a(n) = hyper2F0([-n/2,(1-n)/2],[],2). - _Peter Luschny_, Aug 21 2014
%F 0 = a(n)*(+a(n+1) + a(n+2) - a(n+3)) + a(n+1)*(-a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Aug 22 2014
%F From _Peter Bala_, Oct 06 2021: (Start)
%F a(n+k) == a(n) (mod k) for all n >= 0 and all positive odd integers k.
%F Hence for each odd k, the sequence obtained by taking a(n) modulo k is a periodic sequence and the exact period divides k. For example, taking a(n) modulo 7 gives the purely periodic sequence [1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, ...] of period 7. For similar results see A047974 and A115329. (End)
%e Sequence starts 1, 1, 2, 4, 10, ... because possibilities are {}, {A}, {AB, BA}, {ABC, ACB, BAC, CBA}, {ABCD, ABDC, ACBD, ADCB, BACD, BADC, CBAD, CDAB, DBCA, DCBA}. - _Henry Bottomley_, Jan 16 2001
%e G.f. = 1 + x + 2*x^2 + 4*x^4 + 10*x^5 + 26*x^6 + 76*x^7 + 232*x^8 + 764*x^9 + ...
%e From _Gus Wiseman_, Jan 08 2021: (Start)
%e The a(4) = 10 standard Young tableaux:
%e 1 2 3 4
%e .
%e 1 2 1 3 1 2 3 1 2 4 1 3 4
%e 3 4 2 4 4 3 2
%e .
%e 1 2 1 3 1 4
%e 3 2 2
%e 4 4 3
%e .
%e 1
%e 2
%e 3
%e 4
%e The a(0) = 1 through a(4) = 10 set partitions into singletons or pairs:
%e {} {{1}} {{1,2}} {{1},{2,3}} {{1,2},{3,4}}
%e {{1},{2}} {{1,2},{3}} {{1,3},{2,4}}
%e {{1,3},{2}} {{1,4},{2,3}}
%e {{1},{2},{3}} {{1},{2},{3,4}}
%e {{1},{2,3},{4}}
%e {{1,2},{3},{4}}
%e {{1},{2,4},{3}}
%e {{1,3},{2},{4}}
%e {{1,4},{2},{3}}
%e {{1},{2},{3},{4}}
%e (End)
%p A000085 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else procname(n-1)+(n-1)*procname(n-2); fi; end;
%p with(combstruct):ZL3:=[S,{S=Set(Cycle(Z,card<3))}, labeled]:seq(count(ZL3,size=n),n=0..25); # _Zerinvary Lajos_, Sep 24 2007
%p with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, m>=card))}, labeled]; end: A:=a(2):seq(count(A, size=n), n=0..25); # _Zerinvary Lajos_, Jun 11 2008
%t <<Combinatorica`; [M[Star[n]]]
%t p[0, x] = 1; p[1, x] = x; p[k_, x_] := p[k, x] = x*p[k - 1, x] + (k - 1)*p[k - 2, x]; Table[Sum[CoefficientList[p[n, x], x][[m]], {m, 1, n + 1}], {n, 0, 15}] (* _Roger L. Bagula_, Oct 06 2006 *)
%t With[{nn=30},CoefficientList[Series[Exp[x+x^2/2],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 28 2013 *)
%t a[ n_] := Sum[(2 k - 1)!! Binomial[ n, 2 k], {k, 0, n/2}]; (* _Michael Somos_, Jun 01 2013 *)
%t a[ n_] := If[ n < 0, 0, HypergeometricU[ -n/2, 1/2, -1/2] / (-1/2)^(n/2)]; (* _Michael Somos_, Jun 01 2013 *)
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x + x^2 / 2], {x, 0, n}]]; (* _Michael Somos_, Jun 01 2013 *)
%t Table[(I/Sqrt[2])^n HermiteH[n, -I/Sqrt[2]], {n, 0, 100}] (* _Emanuele Munarini_, Mar 02 2016 *)
%t a[n_] := Sum[StirlingS1[n, k]*2^k*BellB[k, 1/2], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jul 18 2017, after _Emanuele Munarini_ *)
%t RecurrenceTable[{a[n] == a[n-1] + (n-1)*a[n-2], a[0] == 1, a[1] == 1}, a, {n, 0, 20}] (* _Joan Ludevid_, Jun 17 2022 *)
%t sds[{}]:={{}};sds[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sds[Complement[set,s]]]/@Cases[Subsets[set,{1,2}],{i,___}]; Table[Length[sds[Range[n]]],{n,0,10}] (* _Gus Wiseman_, Jan 11 2021 *)
%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( x + x^2 / 2 + x * O(x^n)), n))}; /* _Michael Somos_, Nov 15 2002 */
%o (PARI) N=66; x='x+O('x^N); egf=exp(x+x^2/2); Vec(serlaplace(egf)) \\ _Joerg Arndt_, Mar 07 2013
%o (Haskell) a000085 n = a000085_list !! n
%o a000085_list = 1 : 1 : zipWith (+)
%o (zipWith (*) [1..] a000085_list) (tail a000085_list) -- _Reinhard Zumkeller_, May 16 2013
%o (Maxima) B(n,x):=sum(stirling2(n,k)*x^k,k,0,n);
%o a(n):=sum(stirling1(n,k)*2^k*B(k,1/2),k,0,n);
%o makelist(a(n),n,0,40); /* _Emanuele Munarini_, May 16 2014 */
%o (Maxima) makelist((%i/sqrt(2))^n*hermite(n,-%i/sqrt(2)),n,0,12); /* _Emanuele Munarini_, Mar 02 2016 */
%o (Sage) A000085 = lambda n: hypergeometric([-n/2,(1-n)/2], [], 2)
%o [simplify(A000085(n)) for n in range(28)] # _Peter Luschny_, Aug 21 2014
%o (Sage) def a85(n): return sum(factorial(n) / (factorial(n-2*k) * 2**k * factorial(k)) for k in range(1+n//2))
%o for n in range(100): print(n, a85(n)) # _Manfred Scheucher_, Jan 07 2018
%o (Python)
%o from math import factorial
%o def A000085(n): return sum(factorial(n)//(factorial(n-(k<<1))*factorial(k)*(1<<k)) for k in range((n>>1)+1)) # _Chai Wah Wu_, Aug 31 2023
%Y Cf. A001147, A001470, A053529, A099174, A136281-A136283, A001680, A001681.
%Y See also A005425 for another version of the switchboard problem.
%Y Equals 2 * A001475(n-1) for n>1.
%Y First column of array A099020.
%Y A069943(n+1)/A069944(n+1) = a(n)/A000142(n) in lowest terms.
%Y Cf. A152736, A128229. - _Gary W. Adamson_, Dec 12 2008
%Y Diagonal of A182172. - _Alois P. Heinz_, May 30 2012
%Y Cf. A001813, A006882, A135401, A297708. - _Manfred Scheucher_, Jan 07 2018
%Y Row sums of: A047884, A049403, A096713 (absolute value), A100861, A104556 (absolute value), A111924, A117506 (M_4 numbers), A122848, A238123.
%Y A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
%Y A322661 counts labeled covering half-loop-graphs.
%Y A339742 counts factorizations into distinct primes or squarefree semiprimes.
%Y Cf. A000110, A000258, A000700, A000701, A321729, A321730, A321737.
%K nonn,core,easy,nice
%O 0,3
%A _N. J. A. Sloane_ and _J. H. Conway_