A297708 -id:A297708 - cp's OEIS frontend

A000085 Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, 211799312, 997313824, 4809701440, 23758664096, 119952692896, 618884638912, 3257843882624, 17492190577600, 95680443760576, 532985208200576, 3020676745975552

Offset: 0

N. J. A. Sloane and J. H. Conway

a(n) is also the number of n X n symmetric permutation matrices.
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
a(n) is also the number of independent vertex sets and vertex covers in the n-triangular graph. - Eric W. Weisstein, May 22 2017
Equivalently, this is the number of graphs on n labeled nodes with degrees at most 1. - Don Knuth, Mar 31 2008
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
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
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
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
Row sums of the exponential Riordan array (e^(x^2/2),x). - Paul Barry, Jan 12 2006
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
Equals row sums of triangle A152736 starting with offset 1. - Gary W. Adamson, Dec 12 2008
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
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
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
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
Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013
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
Getu (1991) says these numbers are also known as "telephone numbers". - N. J. A. Sloane, Nov 23 2015
a(n) is the number of elements x in the symmetric group S_n such that x^2 = e where e is the identity. - Jianing Song, Aug 22 2018 [Edited on Jul 24 2025]
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
From Thomas Anton, Apr 20 2020: (Start)
Apparently a(n) = b*c where b is odd iff a(n+b) (when a(n) is defined) is divisible by b.
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)
From Iosif Pinelis, Mar 12 2021: (Start)
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.
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)

			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
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 + ...
From _Gus Wiseman_, Jan 08 2021: (Start)
The a(4) = 10 standard Young tableaux:
  1 2 3 4
.
  1 2   1 3   1 2 3   1 2 4   1 3 4
  3 4   2 4   4       3       2
.
  1 2   1 3   1 4
  3     2     2
  4     4     3
.
  1
  2
  3
  4
The a(0) = 1 through a(4) = 10 set partitions into singletons or pairs:
  {}  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{3,4}}
             {{1},{2}}  {{1,2},{3}}    {{1,3},{2,4}}
                        {{1,3},{2}}    {{1,4},{2,3}}
                        {{1},{2},{3}}  {{1},{2},{3,4}}
                                       {{1},{2,3},{4}}
                                       {{1,2},{3},{4}}
                                       {{1},{2,4},{3}}
                                       {{1,3},{2},{4}}
                                       {{1,4},{2},{3}}
                                       {{1},{2},{3},{4}}
(End)

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 32, 911.
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.
W. Fulton, Young Tableaux, Cambridge, 1997.
D. E. Knuth, The Art of Computer Programming, Vol. 3, Section 5.1.4, p. 65.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
T. Muir, A Treatise on the Theory of Determinants. Dover, NY, 1960, p. 6.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 86.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..800 (first 101 terms from T. D. Noe)
Ahmad Zainy Al-Yasry, Short Steps in Noncommutative Geometry, arXiv:1901.03640 [math.OA], 2019.
T. Amdeberhan and V. H. Moll, Involutions and their progenies, 2014.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem arXiv:0907.0513 [math.CO], 2009.
Joerg Arndt, Generating Random Permutations, PhD thesis, Australian National University, Canberra, Australia, (2010).
Joerg Arndt, Matters Computational (The Fxtbook), pp. 279-280.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
D. Barsky, Analyse p-adique et suites classiques de nombres, Sem. Loth. Comb. B05b (1981) 1-21.
C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
E. A. Bender and S. G. Williamson, Foundations of Combinatorics with Applications (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]
S. Bilotta, E. Pergola, R. Pinzani, and S. Rinaldi, Recurrence relations versus succession rules, arXiv preprint arXiv:1301.2967 [cs.DM], 2013. - From _N. J. A. Sloane_, Feb 12 2013
S. Bilotta, E. Pergola, R. Pinzani, and S. Rinaldi, Recurrence Relations, Succession Rules, and the Positivity Problem, 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.
Aram Bingham and Özlem Uğurlu, Sects, rooks, pyramids, partitions and paths for type DIII clans, arXiv:1907.08875 [math.CO], 2019.
Anders Björner and Richard P. Stanley, A combinatorial miscellany, 2010.
P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering, arXiv:quant-ph/0405103, 2004-2006. The title of this paper in the arXiv was later changed to "Some useful combinatorial formulas for bosonic operators"
P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Some useful combinatorial formulas for bosonic operators, J. Math. Phys. 46, 052110 (2005) (6 pages).
Tobias Boege and Thomas Kahle, Construction Methods for Gaussoids, arXiv:1902.11260 [math.CO], 2019.
D. Bundala, M. Codish, L. Cruz-Filipe et al., Optimal-Depth Sorting Networks, arXiv preprint arXiv:1412.5302 [cs.DS], 2014.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
D. Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.
C.-O. Chow, Counting involutory, unimodal and alternating signed permutations, Discr. Math. 306 (2006), 2222-2228.
S. Chowla, and I.N. Hernstein, W.K. Moore, On recursions connected with symmetric groups. I, Canadian Journal of Mathematics, 3 (1951), 328-334.
M. Codish, L. Cruz-Filipe and P. Schneider-Kamp, The Quest for Optimal Sorting Networks: Efficient Generation of Two-Layer Prefixes, arXiv preprint arXiv:1404.0948 [cs.DS], 2014.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv preprint arXiv:1210.6908 [math.CO], 2012.
I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014.
I. Dolinka, J. East, and R. D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015-2016.
R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
S. Dulucq and J.-G. Penaud, Cordes, arbres et permutations, Discrete Math. 117 (1993), no. 1-3, 89-105.
Shalosh B. Ekhad, Manuel Kauers, and Doron Zeilberger, The O(1/n^85) Asymptotic expansion of OEIS sequence A85, arxiv preprint arXiv:2410.16334 [math.CO], October 2024.
Shalosh B. Ekhad and Doron Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux
Shalosh B. Ekhad and Doron Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, [Local copy, pdf file only, no active links]
Shalosh B. Ekhad and Doron Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, 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]
Shalosh B. Ekhad, Manuel Kauers, and Doron Zeilberger, The O(1/n^85) Asymptotic expansion of OEIS sequence A85, arXiv:2410.16334 [math.CO], 2024.
FindStat - Combinatorial Statistic Finder, Standard Young tableaux
Amy M. Fu and Frank Z. K. Li, Joint Distributions of Permutation Statistics and the Parabolic Cylinder Functions, arXiv:1809.07465 [math.CO], 2018.
Mohammad Ganjtabesh, Armin Morabbi and Jean-Marc Steyaert, Enumerating the number of RNA structures
S. Garrabrant and I. Pak, Counting with irrational tiles, arXiv:1407.8222 [math.CO], 2014.
S. Getu, Evaluating determinants via generating functions, Mathematics Magazine 64 (1991), 45-53.
J. Gilder, Rooks inviolate revisited II, Math. Gaz., 70 (1986), 44-45.
A. M. Goyt, Avoidance of partitions of a 3-element set, arXiv:math/0603481 [math.CO], 2006.
Greenhill, Catherine; McKay, Brendan D., Counting loopy graphs with given degree, Linear Algebra Appl. 436, No. 4, 901-926 (2012)
H. Gupta, Enumeration of symmetric matrices, Duke Math. J., 35 (1968), vol 3, 653-659
H. Gupta, Enumeration of symmetric matrices (annotated scanned copy)
C. Gutschwager, Reduced Kronecker products which are multiplicity free or contain only a few components, Eur. J. Combinat. 31 (2010) 1996-2005 doi:10.1016/j.ejc.2010.05.008, Remark 2.13.
T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv preprint arXiv:1302.6150 [math.RT], 2013.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 17
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 22
András Kaszanyitzky, The GraftalLace Cellular Automaton, arXiv:1805.11532 [nlin.CG], 2018.
Manuel Kauers, Doron Zeilberger, Counting Standard Young Tableaux With Restricted Runs, arXiv:2006.10205 [math.CO], 2020.
Dongsu Kim and Jang Soo Kim, A combinatorial approach to the power of 2 in the number of involutions, J. Comb. Theory Ser. A 117 (8) (2010): 1082-1094
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 227.
P. L. Krapivsky and J. M. Luck, Coverage fluctuations in theater models, arXiv:1902.04365 [cond-mat.stat-mech], 2019.
M. B. Kutler and C. R. Vinroot, On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups, JIS 13 (2010) #10.3.6.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Steven Linton, James Propp, Tom Roby, and Julian West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
T. Mansour and M. Shattuck, Partial matchings and pattern avoidance, Appl. Anal. Discrete Math. (to appear, 2012); - From _N. J. A. Sloane_, Jan 04 2013
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1.
F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
T. Mueller, Finite group actions and asymptotic expansion of e^P(z), Combinatorica, 17 (4) (1997), 523-554.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
I. Pak, G. Panova, and E. Vallejo, Kronecker products, characters, partitions, and the tensor square conjectures, arXiv:1304.0738 [math.CO], 2013.
D. Panyushev, On the orbits of a Borel subgroup in abelian ideals, arXiv preprint arXiv:1407.6857 [math.AG], 2014.
P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
R. Petuchovas, Asymptotic analysis of the cyclic structure of permutations, arXiv:1611.02934 [math.CO], p. 6, 2016.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
J. L. Ramírez and G. N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014).
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
R. W. Robinson, Counting arrangements of bishops, Lect. Notes Math. 560 (1976), 198-214. See D_n.
H. A. Rothe, Ueber Permutationen, in Beziehung auf die Stellen ihrer Elemente. Anwendung der daraus abgeleiteten Sätze auf das Eliminationsproblem, in Carl Hindenberg, ed., Sammlung Combinatorisch-Analytischer Abhandlungen, pp. 263-305, Bey G. Fleischer dem jüngern, 1800; see p. 282.
Ahmad Sabri and Vincent Vajnovszki, Exhaustive generation for ballot sequences in lexicographic and Gray code order, CEUR Workshop Proceedings, 2018.
Ahmad Sabri and Vincent Vajnovszki, On the exhaustive generation of generalized ballot sequences in lexicographic and Gray code order, Pure Mathematics and Applications (2019) Vol. 28, Issue 1, 109-119.
A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, Combinatorial physics, normal order and model Feynman graphs, arXiv:quant-ph/0310174, 2003.
Bridget Eileen Tenner, Star factorizations and noncrossing partitions, arXiv:2004.03286 [math.CO], 2020.
Robert F. Tichy and Stephan Wagner, Extremal Problems for Topological Indices in Combinatorial Chemistry, Journal of Computational Biology. September 2005, 12(7): 1004-1013.
Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Hosoya Index
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Permutation Involution
Eric Weisstein's World of Mathematics, Triangular Graph
Eric Weisstein's World of Mathematics, Vertex Cover
Eric Weisstein's World of Mathematics, Young Tableau
Allan Wentink, Symmetry with Independent Blocks, 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]
Wikipedia, Telephone numbers.
Wikipedia, Young tableau.
Index entries for "core" sequences
Index entries for sequences related to Young tableaux.
Index entries for related partition-counting sequences

Cf. A001147, A001470, A053529, A099174, A136281-A136283, A001680, A001681.
See also A005425 for another version of the switchboard problem.
Equals 2 * A001475(n-1) for n>1.
First column of array A099020.
A069943(n+1)/A069944(n+1) = a(n)/A000142(n) in lowest terms.
Cf. A152736, A128229. - Gary W. Adamson, Dec 12 2008
Diagonal of A182172. - Alois P. Heinz, May 30 2012
Cf. A001813, A006882, A135401, A297708. - Manfred Scheucher, Jan 07 2018
Row sums of: A047884, A049403, A096713 (absolute value), A100861, A104556 (absolute value), A111924, A117506 (M_4 numbers), A122848, A238123.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Cf. A000110, A000258, A000700, A000701, A321729, A321730, A321737.

a000085 n = a000085_list !! n
  a000085_list = 1 : 1 : zipWith (+)
    (zipWith (*) [1..] a000085_list) (tail a000085_list) -- Reinhard Zumkeller, May 16 2013

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;
with(combstruct):ZL3:=[S,{S=Set(Cycle(Z,card<3))}, labeled]:seq(count(ZL3,size=n),n=0..25); # Zerinvary Lajos, Sep 24 2007
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

<Roger L. Bagula, Oct 06 2006 *)
With[{nn=30},CoefficientList[Series[Exp[x+x^2/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 28 2013 *)
a[ n_] := Sum[(2 k - 1)!! Binomial[ n, 2 k], {k, 0, n/2}]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := If[ n < 0, 0, HypergeometricU[ -n/2, 1/2, -1/2] / (-1/2)^(n/2)]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x + x^2 / 2], {x, 0, n}]]; (* Michael Somos, Jun 01 2013 *)
Table[(I/Sqrt[2])^n HermiteH[n, -I/Sqrt[2]], {n, 0, 100}] (* Emanuele Munarini, Mar 02 2016 *)
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 *)
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 *)
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 *)

B(n,x):=sum(stirling2(n,k)*x^k,k,0,n);
  a(n):=sum(stirling1(n,k)*2^k*B(k,1/2),k,0,n);
  makelist(a(n),n,0,40); /* Emanuele Munarini, May 16 2014 */

makelist((%i/sqrt(2))^n*hermite(n,-%i/sqrt(2)),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp( x + x^2 / 2 + x * O(x^n)), n))}; /* Michael Somos, Nov 15 2002 */

N=66; x='x+O('x^N); egf=exp(x+x^2/2); Vec(serlaplace(egf)) \\ Joerg Arndt, Mar 07 2013

from math import factorial
def A000085(n): return sum(factorial(n)//(factorial(n-(k<<1))*factorial(k)*(1<>1)+1)) # Chai Wah Wu, Aug 31 2023

A000085 = lambda n: hypergeometric([-n/2,(1-n)/2], [], 2)
[simplify(A000085(n)) for n in range(28)] # Peter Luschny, Aug 21 2014

def a85(n): return sum(factorial(n) / (factorial(n-2*k) * 2**k * factorial(k)) for k in range(1+n//2))
for n in range(100): print(n, a85(n)) # Manfred Scheucher, Jan 07 2018

D-finite with recurrence a(0) = a(1) = 1, a(n) = a(n-1) + (n-1)*a(n-2) for n>1.
E.g.f.: exp(x+x^2/2).
a(n) = a(n-1) + A013989(n-2) = A013989(n)/(n+1) = 1+A001189(n).
a(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)!*2^k*k!).
a(m+n) = Sum_{k>=0} k!*binomial(m, k)*binomial(n, k)*a(m-k)*a(n-k). - Philippe Deléham, Mar 05 2004
For n>1, a(n) = 2*(A000900(n) + A000902(floor(n/2))). - Max Alekseyev, Oct 31 2015
The e.g.f. y(x) satisfies y^2 = y''y' - (y')^2.
a(n) ~ c*(n/e)^(n/2)exp(n^(1/2)) where c=2^(-1/2)exp(-1/4). [Chowla]
a(n) = HermiteH(n, 1/(sqrt(2)*i))/(-sqrt(2)*i)^n, where HermiteH are the Hermite polynomials. - Karol A. Penson, May 16 2002
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006
For asymptotics see the Robinson paper.
a(n) = Sum_{m=0..n} A099174(n,m). - Roger L. Bagula, Oct 06 2006
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
From Gary W. Adamson, Dec 29 2008: (Start)
a(n) = (n-1)*a(n-2) + a(n-1); e.g., a(7) = 232 = 6*26 + 76.
Starting with offset 1 = eigensequence of triangle A128229. (End)
a(n) = (1/sqrt(2*Pi))*Integral_{x=-oo..oo} exp(-x^2/2)*(x+1)^n. - Groux Roland, Mar 14 2011
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
From Sergei N. Gladkovskii, Dec 03 2011 - Oct 28 2013: (Start)
Continued fractions:
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)).
G.f.: 1/(U(0) - x) where U(k) = 1 + x*(k+1) - x*(k+1)/(1 - x/U(k+1)).
G.f.: 1/Q(0) where Q(k) = 1 + x*k - x/(1 - x*(k+1)/Q(k+1)).
G.f.: -1/(x*Q(0)) where Q(k) = 1 - 1/x - (k+1)/Q(k+1).
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)
a(n) ~ (1/sqrt(2)) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 07 2014
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
a(n) = hyper2F0([-n/2,(1-n)/2],[],2). - Peter Luschny, Aug 21 2014
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
From Peter Bala, Oct 06 2021: (Start)
a(n+k) == a(n) (mod k) for all n >= 0 and all positive odd integers k.
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)

A006882 Double factorials n!!: a(n) = n*a(n-2) for n > 1, a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425, 185794560, 654729075, 3715891200, 13749310575, 81749606400, 316234143225, 1961990553600, 7905853580625, 51011754393600, 213458046676875, 1428329123020800

Offset: 0

Robert Munafo

Product of pairs of successive terms gives factorials in increasing order. - Amarnath Murthy, Oct 17 2002
a(n) = number of down-up permutations on [n+1] for which the entries in the even positions are increasing. For example, a(3)=3 counts 2143, 3142, 4132. Also, a(n) = number of down-up permutations on [n+2] for which the entries in the odd positions are decreasing. For example, a(3)=3 counts 51423, 52413, 53412. - David Callan, Nov 29 2007
The double factorial of a positive integer n is the product of the positive integers <= n that have the same parity as n. - Peter Luschny, Jun 23 2011
For n even, a(n) is the number of ways to place n points on an n X n grid with pairwise distinct abscissa, pairwise distinct ordinate, and 180-degree rotational symmetry. For n odd, the number of ways is a(n-1) because the center point can be considered "fixed". For 90-degree rotational symmetry cf. A001813, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017
Could be extended to include a(-1) = 1. But a(-2) is not defined, otherwise we would have 1 = a(0) = 0*a(-2). - Jianing Song, Oct 23 2019

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 15*x^5 + 48*x^6 + 105*x^7 + 384*x^8 + ...

Putnam Contest, 4 Dec. 2004, Problem A3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 0..806 (terms 0..100 from T. D. Noe)
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.
CombOS - Combinatorial Object Server, Generate colored permutations
Joseph E. Cooper III, A recurrence for an expression involving double factorials, arXiv:1510.00399 [math.CO], 2015.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Gary T. Leavens and Mike Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
Peter Luschny, Multifactorials
B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425-426.
Rudolph Ondrejka, Tables of double factorials, Math. Comp., Vol. 24, No. 109 (1970), p. 231.
Eric Weisstein's World of Mathematics, Double Factorial.
Eric Weisstein's World of Mathematics, Multifactorial.
Index entries for sequences related to factorial numbers
Index entries for "core" sequences

Bisections are A000165 and A001147. These two entries have more information.
Cf. A052319, A143280.
A diagonal of A202212.
Cf. A000085, A001813, A135401, A297708. - Manfred Scheucher, Jan 07 2018

a006882 n = a006882_list !! n
a006882_list = 1 : 1 : zipWith (*) [2..] a006882_list
-- Reinhard Zumkeller, Oct 23 2014

DoubleFactorial:=func< n | &*[n..2 by -2] >; [ DoubleFactorial(n): n in [0..28] ]; // Klaus Brockhaus, Jan 23 2011

A006882 := proc(n) option remember; if n <= 1 then 1 else n*A006882(n-2); fi; end;
A006882 := proc(n) doublefactorial(n) ; end proc; seq(A006882(n),n=0..10) ; # R. J. Mathar, Oct 20 2009
A006882 := n -> mul(k, k = select(k -> k mod 2 = n mod 2, [$1 .. n])):  seq(A006882(n), n = 0 .. 10); # Peter Luschny, Jun 23 2011
A006882 := proc(n) if n=0 then 1 else mul(n-2*k, k=0..floor(n/2)-1); fi; end; # N. J. A. Sloane, May 27 2016

Array[ #!!&, 40, 0 ]
multiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*multiFactorial[n - k, k]]]; Array[ multiFactorial[#, 2] &, 27, 0] (* Robert G. Wilson v, Apr 23 2011 *)

{a(n) = prod(i=0, (n-1)\2, n - 2*i )} \\ Improved by M. F. Hasler, Nov 30 2013

{a(n) = if( n<2, n>=0, n * a(n-2))}; /* Michael Somos, Apr 06 2003 */

{a(n) = if( n<0, 0, my(E); E = exp(x^2 / 2 + x * O(x^n)); n! * polcoeff( 1 + E * x * (1 + intformal(1 / E)), n))}; /* Michael Somos, Apr 06 2003 */

from sympy import factorial2
def A006882(n): return factorial2(n) # Chai Wah Wu, Apr 03 2021

a(n) = Product_{i=0..floor((n-1)/2)} (n - 2*i).
E.g.f.: 1+exp(x^2/2)*x*(1+sqrt(Pi/2)*erf(x/sqrt(2))). - Wouter Meeussen, Mar 08 2001
Satisfies a(n+3)*a(n) - a(n+1)*a(n+2) = (n+1)!. [Putnam Contest]
a(n) = n!/a(n-1). - Vaclav Kotesovec, Sep 17 2012
a(n) * a(n+3) = a(n+1) * (a(n+2) + a(n)). a(n) * a(n+1) = (n+1)!. - Michael Somos, Dec 29 2012
a(n) ~ c * n^((n+1)/2) / exp(n/2), where c = sqrt(Pi) if n is even, and c = sqrt(2) if n is odd. - Vaclav Kotesovec, Nov 08 2014
a(2*n) = 2^n*a(n)*a(n-1). a(2^n) = 2^(2^n - 1) * 1!! * 3!! * 7!! * ... * (2^(n-1) - 1)!!. - Peter Bala, Nov 01 2016
a(n) = 2^h*(2/Pi)^(sin(Pi*h)^2/2)*Gamma(h+1) where h = n/2. This analytical extension supports the view that a(-1) = 1 is a meaningful numerical extension. With this definition (-1/2)!! = Gamma(3/4)/Pi^(1/4). - Peter Luschny, Oct 24 2019
a(n) ~ (n+1/6)*sqrt((2/e)*(n/e)^(n-1)*(Pi/2)^(cos(n*Pi/2)^2)). - Peter Luschny, Oct 25 2019
Sum_{n>=0} 1/a(n) = A143280. - Amiram Eldar, Nov 10 2020
Sum_{n>=0} 1/(a(n)*a(n+1)) = e - 1. - Andrés Ventas, Apr 12 2021

A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

Original entry on oeis.org

1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000, 830034394580628357120000, 58102407620643984998400000

Offset: 0

N. J. A. Sloane

Counts binary rooted trees (with out-degree <= 2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.
Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003
Number of broadcast schemes in the complete graph on n+1 vertices, K_{n+1}. - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
Hankel transform is A137565. - Paul Barry, Nov 25 2009
The e.g.f. of 1/a(n) = n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. - Wolfdieter Lang, Jan 09 2012
From Tom Copeland, Nov 15 2014: (Start)
Aerated with intervening zeros (1,0,2,0,12,0,120,...) = a(n) (cf. A123023 and A001147), the e.g.f. is e^(t^2), so this is the base for the Appell sequence with e.g.f. e^(t^2) e^(x*t) = exp(P(.,x),t) (reverse A059344, cf. A099174, A066325 also). P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for e^(-t^2)e^(x*t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), e.g., (P(.,t))^n = P(n,t).
Equals A000407*2 with leading 1 added. (End)
a(n) is also the number of square roots of any permutation in S_{4*n} whose disjoint cycle decomposition consists of 2*n transpositions. - Luis Manuel Rivera Martínez, Mar 04 2015
Self-convolution gives A076729. - Vladimir Reshetnikov, Oct 11 2016
For n > 1, it follows from the formula dated Aug 07 2013 that a(n) is a Zumkeller number (A083207). - Ivan N. Ianakiev, Feb 28 2017
For n divisible by 4, a(n/4) is the number of ways to place n points on an n X n grid with pairwise distinct abscissae, pairwise distinct ordinates, and 90-degree rotational symmetry. For n == 1 (mod 4), the number of ways is a((n-1)/4) because the center point can be considered "fixed". For 180-degree rotational symmetry see A006882, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017

			The following permutations of order 8 and their reversals have this property:
  1 7 3 5 2 4 0 6
  1 7 4 2 5 3 0 6
  2 3 7 6 1 0 4 5
  2 4 7 1 6 0 3 5
  3 2 6 7 0 1 5 4
  3 5 1 7 0 6 2 4

D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.6, Eq. 32.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
Eugene McDonnell, "Magic Squares and Permutations" APL Quote-Quad 7.3 (Fall, 1976)
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
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).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Rosa Maria Pidatella, Hermite and Laguerre Functions: a Unifying Point of View, Università degli Studi di Catania, Sicily, Italy (2019).
Murray Bremner and Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - _N. J. A. Sloane_, Apr 18 2014
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Elliot J. Carr and Matthew J. Simpson, New homogenization approaches for stochastic transport through heterogeneous media, arXiv:1810.08890 [physics.bio-ph], 2018.
W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, Parity reversing involutions on plane trees and 2-Motzkin paths, arXiv:math/0503300 [math.CO], 2005.
Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D14 (1976), 1536-1553.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 8.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 127
S. Goodenough and C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.
S. Goodenough and C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16,
H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 115
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin. 52 (2012), 41-54 (Theorem 1).
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
R. J. Marsh and P. P. Martin, Pascal arrays: counting Catalan sets, arXiv:math/0612572 [math.CO], 2006.
Calin D. Morosan, On the number of broadcast schemes in networks, Information Processing Letters, Volume 100, Issue 5 (2006), 188-193.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Index to divisibility sequences
Index entries for related partition-counting sequences

Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially the same sequence), A000165, A047055, A047657, A084947, A084948, A084949, A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.
Catalan(n-1)*M^(n-1)*n! for M=1,2,3,4,5,6: A001813, A052714 (or A144828), A221954, A052734, A221953, A221955.
Cf. A123023, A059344, A099174, A066325, A001700, A000407, A006882.
Cf. A000085, A135401, A297708. - Manfred Scheucher, Jan 07 2018

List([0..20],n->Factorial(2*n)/Factorial(n)); # Muniru A Asiru, Nov 01 2018

[Factorial(2*n)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 09 2018

A001813 := n->(2*n)!/n!;
A001813 := n -> mul(k, k = select(k-> k mod 4 = 2,[$1 .. 4*n])):
seq(A001813(n), n=0..16);  # Peter Luschny, Jun 23 2011

Table[(2n)!/n!, {n,0,20}] (* Harvey P. Dale, May 02 2011 *)

makelist(binomial(n+n, n)*n!,n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=binomial(n+n,n)*n! \\ Charles R Greathouse IV, Jun 15 2011

first(n) = x='x+O('x^n); Vec(serlaplace((1 - 4*x)^(-1/2))) \\ Iain Fox, Jan 01 2018 (corrected by Iain Fox, Jan 11 2018)

from math import factorial
def A001813(n): return factorial(n<<1)//factorial(n) # Chai Wah Wu, Feb 14 2023

[binomial(2*n,n)*factorial(n) for n in range(0, 17)] # Zerinvary Lajos, Dec 03 2009

E.g.f.: (1-4*x)^(-1/2).
a(n) = (2*n)!/n! = Product_{k=0..n-1} (4*k + 2) = A081125(2*n).
Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)) dx, n >= 0. This representation is unique. - Karol A. Penson, Sep 18 2001
Define a'(1)=1, a'(n) = Sum_{k=1..n-1} a'(n-k)*a'(k)*C(n, k); then a(n)=a'(n+1). - Benoit Cloitre, Apr 27 2003
With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - Paul Barry, May 09 2003
a(n) = A000680(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (4*i + 2) = 4^n*Pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
For asymptotics, see the Robinson paper.
a(k) = (2*k)!/k! = Sum_{i=1..k+1} |A008275(i,k+1)| * k^(i-1). - André F. Labossière, Jun 21 2007
a(n) = 12*A051618(a) n >= 2. - Zerinvary Lajos, Feb 15 2008
a(n) = A000984(n)*A000142(n). - Zerinvary Lajos, Mar 25 2008
a(n) = A016825(n-1)*a(n-1). - Roger L. Bagula, Sep 17 2008
a(n) = (-1)^n*A097388(n). - D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
From Paul Barry, Jan 15 2009: (Start)
G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-... (continued fraction);
a(n) = (n+1)!*A000108(n). (End)
a(n) = Sum_{k=0..n} A132393(n,k)*2^(2n-k). - Philippe Deléham, Feb 10 2009
G.f.: 1/(1-2x-8x^2/(1-10x-48x^2/(1-18x-120x^2/(1-26x-224x^2/(1-34x-360x^2/(1-42x-528x^2/(1-... (continued fraction). - Paul Barry, Nov 25 2009
a(n) = A173333(2*n,n) for n>0; cf. A006963, A001761. - Reinhard Zumkeller, Feb 19 2010
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
  2, 2, 0, 0, 0, 0, ...
  4, 4, 4, 0, 0, 0, ...
  6, 6, 6, 6, 0, 0, ...
  8, 8, 8, 8, 8, 0, ...
  ...
(End)
a(n) = (-2)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0), where Q(k) = 1 + x*(4*k+2) - x*(4*k+4)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 8*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + 1/(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
D-finite with recurrence: a(n) = (4*n-6)*a(n-2) + (4*n-3)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 07 2013
Sum_{n>=0} 1/a(n) = (exp(1/4)*sqrt(Pi)*erf(1/2) + 2)/2 = 1 + A214869, where erf(x) is the error function. - Ilya Gutkovskiy, Nov 10 2016
Sum_{n>=0} (-1)^n/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)), where erfi(x) is the imaginary error function. - Amiram Eldar, Feb 20 2021
a(n) = 1/([x^n] hypergeom([1], [1/2], x/4)). - Peter Luschny, Sep 13 2024
a(n) = 2^n*n!*JacobiP(n, -1/2, -n, 3). - Peter Luschny, Jan 22 2025
G.f.: 2F0(1,1/2;;4x). - R. J. Mathar, Jun 07 2025

More terms from James Sellers, May 01 2000

a(n) is also the number of ways to place n points on an n X n grid with pairwise distinct abscissa, pairwise distinct ordinate, mirror symmetry and 180-degree rotational symmetry. Note that both diagonals are actually axes of symmetry. See also A297708, A000085, A001813, and A006882. - Manfred Scheucher, Jan 04 2018

a(n) is the number of standard Young tableaux of size n invariant under Schützenberger involution. - Ludovic Schwob, Feb 17 2024

cp's OEIS Frontend

A000085 Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Maxima

Maxima

PARI

PARI

Python

Sage

Sage

Formula

A006882 Double factorials n!!: a(n) = n*a(n-2) for n > 1, a(0) = a(1) = 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

Maxima

PARI

PARI

Python

Sage

Formula

Extensions

A135401 a(n) = number of permutations (p(1),p(2),p(3),...,p(n)) of (1,2,3,...n) each of which is its own inverse and is such that p(k) = n + 1 - p(n+1-k) for all k in the range 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

Extensions