A013989 -id:A013989 - 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)

A001475 a(n) = a(n-1) + n * a(n-2), where a(1) = 1, a(2) = 2.

Original entry on oeis.org

1, 2, 5, 13, 38, 116, 382, 1310, 4748, 17848, 70076, 284252, 1195240, 5174768, 23103368, 105899656, 498656912, 2404850720, 11879332048, 59976346448, 309442319456, 1628921941312, 8746095288800, 47840221880288, 266492604100288, 1510338372987776

Offset: 1

N. J. A. Sloane

nonn

a(n) is the number of set partitions of [n] in which the block containing 1 is of length <= 3 and all other blocks are of length <= 2. Example: a(4)=13 counts all 15 partitions of [4] except 1234 and 1/234. - David Callan, Jul 22 2008

Empirical: a(n) is the sum of the entries in the second-last row of the lower-triangular matrix of coefficients giving the expansion of degree-(n+1) complete homogeneous symmetric functions in the Schur basis of the algebra of symmetric functions. - John M. Campbell, Mar 18 2018

			G.f. = x + 2*x + 5*x^2 + 13*x^3 + 38*x^4 + 116*x^5 + 382*x^6 + 1310*x^7 + ... - _Michael Somos_, Jan 23 2018

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 86 (divided by 2).
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).

John Cerkan, Table of n, a(n) for n = 1..795
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.

Cf. A000085, A001189, A013989, A076276, A248475.

a:=[1, 2];; for n in [3..10^2] do a[n] := a[n-1] + n*a[n-2]; od; a;  # Muniru A Asiru, Jan 25 2018

I:=[1,2]; [n le 2 select I[n] else Self(n-1)+n*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 31 2018

a := proc(n) option remember: if n = 1 then 1 elif n = 2 then 2 elif  n >= 3 then procname(n-1) +n*procname(n-2) fi; end:
seq(a(n), n = 1..100); # Muniru A Asiru, Jan 25 2018

RecurrenceTable[{a[1]==1,a[2]==2,a[n]==a[n-1]+n a[n-2]},a,{n,30}] (* Harvey P. Dale, Apr 21 2012 *)
(* Programs from Michael Somos, Jan 23 2018 *)
a[n_]:= With[{m=n+1}, If[m<2, 0, Sum[(2 k-1)!! Binomial[m, 2 k], {k, 0, m/2}]/2]];
a[n_]:= With[{m=n+1}, If[m<2, 0, HypergeometricU[-m/2, 1/2, -1/2] / (-1/2)^(m/2)/2]];
a[n_]:= With[{m=n+1}, If[m<2, 0, HypergeometricPFQ[{-m/2, (1-m)/2}, {}, 2]/2]];
a[n_]:= If[ n<1, 0, n! SeriesCoefficient[Exp[x+x^2/2]*(1+x)/2, {x, 0, n}]]; (* End *)
Fold[Append[#1, #1[[-1]] + #2 #1[[-2]]] &, {1, 2}, Range[3, 26]] (* Michael De Vlieger, Jan 23 2018 *)

{a(n) = if( n<1, 0, n! * polcoeff( exp( x + x^2/2 + x * O(x^n)) * (1 + x) / 2, n))}; /* Michael Somos, Jan 23 2018 */

my(N=30,x='x+O('x^N)); Vec(serlaplace((1/2)*( (1+x)*exp(x + x^2/2) - 1))) \\ Joerg Arndt, Sep 04 2023

def A001475_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( ((1+x)*exp(x+x^2/2) -1)/2 ).egf_to_ogf().list()
a=A001475_list(40); a[1:] # G. C. Greubel, Sep 03 2023

a(n) = (1/2)*A000085(n+1).
E.g.f.: (1/2)*( (1+x)*exp(x + x^2/2) - 1). - Vladeta Jovovic, Nov 04 2003
Given e.g.f. y(x), then 0 = y'(x) * (1+x) - (y(x)+1/2) * (2+2*x+x^2) = 1 - y''(x) + y'(x)*(1 + x) + 2*y(x). - Michael Somos, Jan 23 2018
0 = +a(n)*(+a(n+1) +a(n+2) -a(n+3)) +a(n+1)*(-a(n+1) +a(n+2)) for all n>0. - Michael Somos, Jan 23 2018
a(n) ~ n^((n+1)/2) / (2^(3/2) * exp(n/2 - sqrt(n) + 1/4)) * (1 + 19/(24*sqrt(n))). - Vaclav Kotesovec, Apr 01 2018

More terms from Harvey P. Dale, Apr 21 2012

A162970 Number of 2-cycles in all involutions of {1,2,...,n}.

Original entry on oeis.org

0, 1, 3, 12, 40, 150, 546, 2128, 8352, 34380, 144100, 626736, 2784288, 12753832, 59692920, 286857600, 1407536896, 7069630608, 36217682352, 189489626560, 1010037302400, 5488251406176, 30348031302688, 170812160339712

Offset: 1

Emeric Deutsch, Jul 22 2009

nonn

			a(3) = 3 because in (1)(2)(3), (1)(23), (12)(3), (13)(2) we have three 2-cycles.

Alois P. Heinz, Table of n, a(n) for n = 1..300

Cf. A000085, A013989.

a[1] := 0: a[2] := 1: for n from 3 to 27 do a[n] := n*(a[n-1]+(n-1)*a[n-2])/(n-2) end do: seq(a[n], n = 1 .. 27);

Range[0, 20]! CoefficientList[ Series[x^2/2  Exp[x+x^2/2], {x, 0, 20}], x] // Rest

a(n) = (1/2)*n*(n-1)*I(n-2) for n>=2, where I(n)=A000085(n) is the number of involutions of {1,2,...,n}.
Rec. rel.: a(n) = [n/(n-2)][a(n-1) + (n-1)a(n-2)], a(1)=0, a(2)=1.
E.g.f.: x^2/2 * exp(x+x^2/2).
a(n) ~ sqrt(2)/4 * n^(n/2+1)*exp(sqrt(n)-n/2-1/4) * (1-17/(24*sqrt(n))). - Vaclav Kotesovec, Aug 15 2013

A099020 Euler-Seidel matrix T(k,n) with start sequence A001147, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 2, 1, 1, 4, 2, 1, 0, 10, 6, 4, 3, 3, 26, 16, 10, 6, 3, 0, 76, 50, 34, 24, 18, 15, 15, 232, 156, 106, 72, 48, 30, 15, 0, 764, 532, 376, 270, 198, 150, 120, 105, 105, 2620, 1856, 1324, 948, 678, 480, 330, 210, 105, 0, 9496, 6876, 5020, 3696, 2748, 2070, 1590, 1260, 1050, 945, 945

Offset: 0

Ralf Stephan, Sep 23 2004

In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences, with the first column the inverse binomial transform of the start sequence.

			1,   0,  1,  0,   3,   0,   15, ...
1,   1,  1,  3,   3,  15,   15, ...
2,   2,  4,  6,  18,  30,  120, ...
4,   6, 10, 24,  48, 150,  330, ...
10, 16, 34, 72, 198, 480, 1590, ...

Alois P. Heinz, Rows n = 0..140, flattened
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

First column is A000085, 2nd A013989, main diagonal is in A099021.

T:= proc(k, n) option remember; `if`(k=0, `if`(irem(n, 2)=0,
      doublefactorial(n-1), 0), T(k-1, n) +T(k-1, n+1))
    end:
seq(seq(T(d-n, n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 14 2012

t[0, n_?EvenQ] := (n-1)!!; t[0, n_?OddQ] := 0; t[k_, n_] := t[k, n] = t[k-1, n] + t[k-1, n+1]; Table[t[k-n, n], {k, 0, 10}, {n, 0, k}] // Flatten (* Jean-François Alcover, Dec 10 2012 *)

Recurrence: T(0, 2n) = (2n-1)!!, T(0, 2n+1) = 0, T(k, n) = T(k-1, n) + T(k-1, n+1).

A111062 Triangle T(n, k) = binomial(n, k) * A000085(n-k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 10, 16, 12, 4, 1, 26, 50, 40, 20, 5, 1, 76, 156, 150, 80, 30, 6, 1, 232, 532, 546, 350, 140, 42, 7, 1, 764, 1856, 2128, 1456, 700, 224, 56, 8, 1, 2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1, 9496, 26200, 34380, 27840, 15960, 6552, 2100, 480, 90, 10, 1

Offset: 0

Philippe Deléham, Oct 07 2005

Triangle related to A000085.
Riordan array [exp(x(2+x)/2),x]. - Paul Barry, Nov 05 2008
Array is exp(S+S^2/2) where S is A132440 the infinitesimal generator for Pascal's triangle. T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then partitioning the remaining n-k elements into sets each of size 1 or 2. Cf. A122832. - Peter Bala, May 14 2012
T(n,k) is equal to the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the partial Brauer monoid of degree n. - James East, Aug 17 2015

			Rows begin:
     1;
     1,    1;
     2,    2,    1;
     4,    6,    3,    1;
    10,   16,   12,    4,    1;
    26,   50,   40,   20,    5,    1;
    76,  156,  150,   80,   30,    6,   1;
   232,  532,  546,  350,  140,   42,   7,  1;
   764, 1856, 2128, 1456,  700,  224,  56,  8, 1;
  2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1;
From _Paul Barry_, Apr 23 2009: (Start)
Production matrix is:
  1, 1,
  1, 1, 1,
  0, 2, 1, 1,
  0, 0, 3, 1, 1,
  0, 0, 0, 4, 1, 1,
  0, 0, 0, 0, 5, 1, 1,
  0, 0, 0, 0, 0, 6, 1, 1,
  0, 0, 0, 0, 0, 0, 7, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 8, 1, 1 (End)
From _Peter Bala_, Feb 12 2017: (Start)
The infinitesimal generator has integer entries and begins
  0
  1  0
  1  2  0
  0  3  3  0
  0  0  6  4  0
  0  0  0 10  5  0
  0  0  0  0 15  6  0
  ...
and is the generalized exponential Riordan array [x + x^2/2!,x].(End)

Muniru A Asiru, Table of n, a(n) for n = 0..1325
Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Loughlin, Enumeration of idempotents in diagram semigroups and algebras, arXiv:1408.2021 [math.GR], 2014.
Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Loughlin, Enumeration of idempotents in diagram semigroups and algebras, J. Combin. Theory Ser. A 131 (2015), 119-152.
Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.

Cf. A000085, A005425 (row sums), A007318, A013989, A122832, A132440.

Cf. A099174, A133314, A159834 (inverse matrix).

Flat(List([0..10],n->List([0..n],k->(Factorial(n)/Factorial(k))*Sum([0..n-k],j->Binomial(j,n-k-j)/(Factorial(j)*2^(n-k-j)))))); # Muniru A Asiru, Jun 29 2018

a[n_] := Sum[(2 k - 1)!! Binomial[n, 2 k], {k, 0, n/2}]; Table[Binomial[n, k] a[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 20 2015, after Michael Somos at A000085 *)

def A111062_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+M[n-1,k]+(k+1)*M[n-1,k+1]
    return M
A111062_triangle(9) # Peter Luschny, Sep 19 2012

Sum_{k>=0} T(m, k)*T(n, k)*k! = T(m+n, 0) = A000085(m+n).
Sum_{k=0..n} T(n, k) = A005425(n).
Apparently satisfies T(n,m) = T(n-1,m-1) + T(n-1,m) + (m+1) * T(n-1,m+1). - Franklin T. Adams-Watters, Dec 22 2005 [corrected by Werner Schulte, Feb 12 2025]
T(n,k) = (n!/k!)*Sum_{j=0..n-k} C(j,n-k-j)/(j!*2^(n-k-j)). - Paul Barry, Nov 05 2008
G.f.: 1/(1-xy-x-x^2/(1-xy-x-2x^2/(1-xy-x-3x^2/(1-xy-x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009
T(n,k) = C(n,k)*Sum_{j=0..n-k} C(n-k,j)*(n-k-j-1)!! where m!!=0 if m is even. - James East, Aug 17 2015
From Tom Copeland, Jun 26 2018: (Start)
E.g.f.: exp[t*p.(x)] = exp[t + t^2/2] e^(x*t).
These polynomials (p.(x))^n = p_n(x) are an Appell sequence with the lowering and raising operators L = D and R = x + 1 + D, with D = d/dx, such that L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x), so the formalism of A133314 applies here, giving recursion relations.
The transpose of the production matrix gives a matrix representation of the raising operator R.
exp(D + D^2/2) x^n= e^(D^2/2) (1+x)^n = h_n(1+x) = p_n(x) = (a. + x)^n, with (a.)^n = a_n = A000085(n) and h_n(x) the modified Hermite polynomials of A099174.
A159834 with the e.g.f. exp[-(t + t^2/2)] e^(x*t) gives the matrix inverse for this entry with the umbral inverse polynomials q_n(x), an Appell sequence with the raising operator  x - 1 - D, such that umbrally composed q_n(p.(x)) = x^n = p_n(q.(x)). (End)

Corrected by Franklin T. Adams-Watters, Dec 22 2005

10th row added by James East, Aug 17 2015

A069943 a(n) = numerator(b(n)), where b(1) = b(2) = 1, b(n) = (b(n-1) + b(n-2))/(n-1).

Original entry on oeis.org

1, 1, 1, 2, 5, 13, 19, 29, 191, 131, 1187, 2231, 17519, 71063, 29881, 323423, 2887921, 13237457, 2397389, 15030317, 742458253, 3748521653, 9670072483, 25451905333, 10932619111, 78684575461, 4163946939067, 11799518538967, 136025604432743

Offset: 1

Benoit Cloitre, Apr 27 2002

Sum_{k>=1} b(k) = exp(3/2). More generally if b(1) = b(2) = ... = b(m) = 1 and b(n+m+1) = (1/(n+m))*(b(n+m) + b(n+m-1) + ... + b(n)) then Sum_{k>=1} b(k) = exp(H(m)) where H(m) = Sum_{j=1..m} 1/j is the m-th harmonic number. [Benoit Cloitre and Boris Gourevitch]

G. C. Greubel, Table of n, a(n) for n = 1..840

Cf. A000085, A000142, A013989, A069944.

A013989:= func< n | (&+[Factorial(n)/(2^k*Factorial(n-2*k)*Factorial(k)): k in [0..Floor(n/2)]]) >;
A069944:= func< n | Numerator(A013989(n-1)/Factorial(n)) >;
[A069944(n): n in [1..40]]; // G. C. Greubel, Aug 17 2022

Table[Numerator[n*(-I/Sqrt[2])^(n-1)*HermiteH[n-1, I/Sqrt[2]]/n!], {n, 40}] (* G. C. Greubel, Aug 17 2022 *)
nxt[{n_,a_,b_}]:={n+1,b,(a+b)/n}; NestList[nxt,{2,1,1},30][[;;,2]]//Numerator (* Harvey P. Dale, Feb 02 2025 *)

@CachedFunction
def A013989(n): return n+1 if (n<2) else (n+1)*(A013989(n-1) + n*A013989(n-2))/n
[numerator(A013989(n-1)/factorial(n)) for n in (1..40)] # G. C. Greubel, Aug 17 2022

a(n)/A069944(n) = A000085(n-1)/A000142(n-1) in lowest terms. [Christian G. Bower, Jan 14 2006]
Numerators in the power series of exp(x+x^2/2) (e.g.f. for involutions, cf. A000085). exp(x+x^2/2) = 1 + x + x^2 + 2/3*x^3 + 5/12*x^4 + 13/60*x^5 + 19/180*x^6 + 29/630*x^7 + 191/10080*x^8 + ... [Joerg Arndt, May 10 2008]
a(n) = numerator( A013989(n-1)/n! ). - G. C. Greubel, Aug 17 2022

A069944 a(n) = denominator(b(n)), where b(1) = b(2) = 1, b(n) = (b(n-1) + b(n-2))/(n-1).

Original entry on oeis.org

1, 1, 1, 3, 12, 60, 180, 630, 10080, 18144, 453600, 2494800, 59875200, 778377600, 1089728640, 40864824000, 1307674368000, 22230464256000, 15390321408000, 380140938777600, 76028187755520000, 1596591942865920000

Offset: 1

Benoit Cloitre, Apr 27 2002

Sum_{k >= 1} b(k) = e^(3/2) where e = 2.718... . More generally if b(1) = b(2) = ... = b(m) = 1 and b(n+m+1) = 1/(n+m)*( b(n+m) + b(n+m-1) + ... + b(n) ) then Sum_{k >= 1} b(k) = e^H(m) where H(m) = Sum_{j=1..m} 1/j is the m-th harmonic number (Benoit Cloitre and Boris Gourevitch).

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A000085, A000142, A013989, A069943.

A013989:= func< n | (&+[Factorial(n)/(2^k*Factorial(n-2*k)*Factorial(k)): k in [0..Floor(n/2)]]) >;
A069944:= func< n | Denominator(A013989(n-1)/Factorial(n-1)) >;
[A069944(n): n in [1..30]]; // G. C. Greubel, Aug 17 2022

Table[Denominator[n*(-I/Sqrt[2])^(n-1)*HermiteH[n-1, I/Sqrt[2]]/n!], {n, 30}] (* G. C. Greubel, Aug 17 2022 *)

@CachedFunction
def A013989(n): return n+1 if (n<2) else (n+1)*(A013989(n-1) + n*A013989(n-2))/n
[denominator(A013989(n-1)/factorial(n)) for n in (1..30)] # G. C. Greubel, Aug 17 2022

A069943(n)/a(n) = A000085(n-1)/A000142(n-1) in lowest terms. - Christian G. Bower, Jan 14 2006

a(n) = denominator( A013989(n-1)/n! ). - G. C. Greubel, Aug 17 2022

A210861 Triangle of coefficients of polynomials v(n,x) jointly generated with A210860; see the Formula section.

Original entry on oeis.org

1, 2, 2, 6, 7, 3, 16, 30, 20, 5, 50, 116, 108, 47, 8, 156, 460, 552, 338, 105, 13, 532, 1842, 2692, 2119, 941, 221, 21, 1856, 7532, 13072, 12574, 7216, 2452, 451, 34, 6876, 31600, 63240, 71860, 50525, 22371, 6035, 895, 55, 26200, 135576, 308568

Offset: 1

Clark Kimberling, Mar 28 2012

Row n ends with F(n+1), where F=A000045 (Fibonacci numbers).
Column 1:  A013989
Alternating row sums:  1,0,2,1,3,2,4,3,5,4,...
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2....2
6....7.....3
16...30....20....5
50...116...108...47...8
First three polynomials v(n,x): 1, 2 + 2x, 6 + 7x + 3x^2

Cf. A210860, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210860 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210861 *)

u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+n)*u(n-1,x)+x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.

A233387 Number of labeled star graphs with added edges.

Original entry on oeis.org

32, 185, 1308, 10822, 102176, 1081908, 12681640, 162880256, 2273437392, 34249286656, 553698389888, 9558929197560, 175471796530816, 3412297318315472, 70064350595106336, 1514554957975079008, 34377185731361631680

Offset: 4

Geoffrey Critzer, Feb 02 2014

nonn

Here, a star graph is a tree on n nodes (n>=4) with one specially designated (center) vertex, v of degree n-1.  We are allowed to add edges so that the degree of any node (excepting v) is at most 3 and so that every cycle includes the vertex v with the possible exception of a single cycle of length n-1 through each non-center vertex. We note that anytime edges are added the original "center" node remains specially designated.  a(n) is the number of such connected simple labeled graphs with a specially designated node.
If we don't add any edges we have a star graph and there are n labelings.
If we add exactly one edge then we produce a cycle of length 3 and we no longer have a tree.
If we add the maximum number of edges we get a wheel graph A171005.

			a(4) = 32. There are 4 labelings for the star graph on 4 nodes. There are 12 labelings after we add one edge. There are 12 labelings after we add two edges. There are 4 labelings after we add 3 edges. 4+12+12+4=32.

Cf. A013989 (with appropriate offset) enumerates such graphs where the maximum degree of non-center vertices is restricted to 2.

nn=20; a=x/(1-x)/2+x/2; Drop[Range[0,nn]! CoefficientList[Series[x Exp[a]+x (Log[1/(1-x)]/2+x^2/4+x/2), {x,0,nn}], x], 4]

Ignoring the first 4 terms the e.g.f. is: x*exp(A(x))+ x*(log(1/(1-x))/2 + x^2/4 + x/2) where A(x) = x/(1-x)/2 + x/2.

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

A001475 a(n) = a(n-1) + n * a(n-2), where a(1) = 1, a(2) = 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

Extensions

A162970 Number of 2-cycles in all involutions of {1,2,...,n}.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A099020 Euler-Seidel matrix T(k,n) with start sequence A001147, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A111062 Triangle T(n, k) = binomial(n, k) * A000085(n-k), 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Mathematica

Sage

Formula

Extensions

A069943 a(n) = numerator(b(n)), where b(1) = b(2) = 1, b(n) = (b(n-1) + b(n-2))/(n-1).

Views