A001680 -id:A001680 - 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)

A080510 Triangle read by rows: T(n,k) gives the number of set partitions of {1,...,n} with maximum block length k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 25, 20, 5, 1, 1, 75, 90, 30, 6, 1, 1, 231, 420, 175, 42, 7, 1, 1, 763, 2016, 1015, 280, 56, 8, 1, 1, 2619, 10024, 6111, 1890, 420, 72, 9, 1, 1, 9495, 51640, 38010, 12978, 3150, 600, 90, 10, 1, 1, 35695, 276980, 244035, 91938, 24024, 4950, 825, 110, 11, 1

Offset: 1

Wouter Meeussen, Mar 22 2003

Row sums are A000110 (Bell numbers). Second column is A001189 (Degree n permutations of order exactly 2).
From Peter Luschny, Mar 09 2009: (Start)
Partition product of Product_{j=0..n-1} ((k + 1)*j - 1) and n! at k = -1, summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A036040.
Same partition product with length statistic is A008277.
Diagonal a(A000217) = A000012.
Row sum is A000110. (End)
From Gary W. Adamson, Feb 24 2011: (Start)
Construct an array in which the n-th row is the partition function G(n,k), where G(n,1),...,G(n,6) = A000012, A000085, A001680, A001681, A110038, A148092, with the first few rows
  1,   1,   1,   1,   1,   1,    1, ... = A000012
  1,   2,   4,  10,  26,  76,  232, ... = A000085
  1,   2,   5,  14,  46, 166,  652, ... = A001680
  1,   2,   5,  15,  51, 196,  827, ... = A001681
  1,   2    5   15   52  202   869, ... = A110038
  1,   2,   5   15   52  203   876, ... = A148092
  ...
Rows tend to A000110, the Bell numbers. Taking finite differences from the top, then reorienting, we obtain triangle A080510.
The n-th row of the array is the eigensequence of an infinite lower triangular matrix with n diagonals of Pascal's triangle starting from the right and the rest zeros. (End)

			T(4,3) = 4 since there are 4 set partitions with longest block of length 3: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}} and {{1,2,4},{3}}.
Triangle begins:
  1;
  1,    1;
  1,    3,     1;
  1,    9,     4,    1;
  1,   25,    20,    5,    1;
  1,   75,    90,   30,    6,   1;
  1,  231,   420,  175,   42,   7,  1;
  1,  763,  2016, 1015,  280,  56,  8,  1;
  1, 2619, 10024, 6111, 1890, 420, 72,  9,  1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.
J. Riordan, Letter, 11/23/1970. See second page of letter.

Cf. A080107, A080337, A008277, A178979, A276922, A327884.
Cf. A157396, A157397, A157398, A157399, A157400, A157401, A157402, A157403, A157404, A157405. - Peter Luschny, Mar 09 2009
Cf. A000012, A000085, A001680, A001681, A110038, A148092. - Gary W. Adamson, Feb 24 2011
Columns k=1..10 give: A000012 (for n>0), A001189, A229245, A229246, A229247, A229248, A229249, A229250, A229251, A229252. - Alois P. Heinz, Sep 17 2013
T(2n,n) gives A276961.
Take differences along rows of A229223. - N. J. A. Sloane, Jan 10 2018

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(b(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
T:= (n, k)-> b(n, k) -b(n, k-1):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Apr 20 2012

<< DiscreteMath`NewCombinatorica`; Table[Length/@Split[Sort[Max[Length/@# ]&/@SetPartitions[n]]], {n, 12}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]*n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; T[n_, k_] := b[n, k]-b[n, k-1]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)

E.g.f. for k-th column: exp(exp(x)*GAMMA(k, x)/(k-1)!-1)*(exp(x^k/k!)-1). - Vladeta Jovovic, Feb 04 2005
From Peter Luschny, Mar 09 2009: (Start)
T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n.
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,...,a_n such that
1*a_1 + 2*a_2 + ... + n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*...*a_n!),
f^a = (f_1/1!)^a_1*...*(f_n/n!)^a_n and f_n = Product_{j=0..n-1} (-1) = (-1)^n. (End)
From Ludovic Schwob, Jan 15 2022: (Start)
T(2n,n) = C(2n,n)*(A000110(n)-1/2) for n>0.
T(n,m) = C(n,m)*A000110(n-m) for 2m > n > 0. (End)

A229223 Number G(n,k) of set partitions of {1,...,n} into sets of size at most k; triangle G(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 10, 14, 15, 0, 1, 26, 46, 51, 52, 0, 1, 76, 166, 196, 202, 203, 0, 1, 232, 652, 827, 869, 876, 877, 0, 1, 764, 2780, 3795, 4075, 4131, 4139, 4140, 0, 1, 2620, 12644, 18755, 20645, 21065, 21137, 21146, 21147

Offset: 0

Alois P. Heinz, Sep 16 2013

John Riordan calls these Allied Bell Numbers. - N. J. A. Sloane, Jan 10 2018
G(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n. G(n,k) = G(n,n) = A000110(n) for k>n.
G(n,k) - G(n,k-1) = A080510(n,k).
A column G(n>=0,k) can be generated by a linear recurrence with polynomial coefficients, where the initial terms correspond with A000110, and the coefficients contain constant factors derived from A008279 (cf. recg(k) in the fourth Maple program below). - Georg Fischer, May 19 2021

			G(4,2) = 10: 1/2/3/4, 12/3/4, 13/2/4, 14/2/3, 1/23/4, 1/24/3, 1/2/34, 12/34, 13/24, 14/23.
Triangle G(n,k) begins:
  1;
  0,  1;
  0,  1,   2;
  0,  1,   4,    5;
  0,  1,  10,   14,   15,
  0,  1,  26,   46,   51,   52;
  0,  1,  76,  166,  196,  202,  203;
  0,  1, 232,  652,  827,  869,  876,  877;
  0,  1, 764, 2780, 3795, 4075, 4131, 4139, 4140;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
J. Riordan, Letter, 11/23/1970

Columns k=0-10 give: A000007, A000012, A000085, A001680, A001681, A110038, A148092, A229224, A229225, A229226, A229227.
Main diagonal gives: A000110. Lower diagonal gives: A058692.
Cf. A066223 (G(2n,2)), A229228 (G(2n,n)), A229229 (G(n^2,n)), A227223 (G(2^n,n)).
Cf. A008279, A080510.

G:= proc(n, k) option remember; `if`(n=0, 1, `if`(k<1, 0,
       add(G(n-k*j, k-1) *n!/k!^j/(n-k*j)!/j!, j=0..n/k)))
    end:
seq(seq(G(n, k), k=0..n), n=0..10);
# second Maple program:
G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j,k) od; % fi
    end:
seq(seq(G(n, k), k=0..n), n=0..10);
# third Maple program:
G:= proc(n, k) option remember; `if`(n=0, 1, add(
      G(n-i, k)*binomial(n-1, i-1), i=1..min(n, k)))
    end:
seq(seq(G(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 26 2017
# fourth Maple program (for columns G(n>=0,k)):
init := n -> seq(a(j) = combinat:-bell(j), j=0..n): # A000110
b := (n, k) -> mul((n - j)/(j + 1), j = 0..k-1):
recg := k -> {(k-1)!*(add(j*b(n, j)*a(n-j), j = 1..k) - n*a(n)), init(k-1)}:
column := proc(k, len) local f; f := gfun:-rectoproc(recg(k), a(n), remember):
map(f, [$0..len-1]) end:
seq(print(column(k, 12)), k=1..9); # Georg Fischer, May 19 2021

g[n_, k_] := g[n, k] = If[n == 0, 1, If[k < 1, 0, Sum[g[n - k*j, k - 1] *n!/k!^j/(n - k*j)!/j!, { j, 0, n/k}]]]; Table[Table[g[n, k], { k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

G(0,k) = 1, G(n,k) = 0 for n>0 and k<1, otherwise G(n,k) = Sum_{j=0..floor(n/k)} G(n-k*j,k-1) * n!/(k!^j*(n-k*j)!*j!).
G(n,k) = G(n-1,k) +(n-1)/1 *(G(n-2,k) +(n-2)/2 *(G(n-3,k) +(n-3)/3 *(G(n-4,k) + ... +(n-(k-1))/(k-1) *G(n-k,k)...))).
E.g.f. of column k: exp(Sum_{j=1..k} x^j/j!).

A188416 T(n,k)=Number of (n*k)Xk binary arrays with rows in nonincreasing order, n ones in every column and no more than 3 ones in any row.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 16, 4, 1, 46, 124, 39, 5, 1, 166, 1308, 723, 81, 6, 1, 652, 17443, 21965, 3217, 150, 7, 1, 2780, 282349, 968230, 252493, 11712, 256, 8, 1, 12644, 5396178, 57967660, 33313408, 2181522, 36659, 410, 9, 1, 61136, 119324602, 4505774905

Offset: 1

R. H. Hardin Mar 30 2011

Table starts
.1..2....5.....14........46..........166............652............2780
.1..3...16....124......1308........17443.........282349.........5396178
.1..4...39....723.....21965.......968230.......57967660......4505774905
.1..5...81...3217....252493.....33313408.....6720357785...1943505834628
.1..6..150..11712...2181522....793369090...498407294632.497114063711190
.1..7..256..36659..15083334..14100262412.25784105550034
.1..8..410.101829..87171345.197263686098
.1..9..625.256901.434626896
.1.10..915.598561
.1.11.1296

			All solutions for 6X2
..1..1....1..1....1..0....1..1
..1..0....1..1....1..0....1..1
..1..0....1..1....1..0....1..0
..0..1....0..0....0..1....0..1
..0..1....0..0....0..1....0..0
..0..0....0..0....0..1....0..0

R. H. Hardin, Table of n, a(n) for n = 1..80

Column 3 is A011863(n+1)

Row 1 is A001680

A144385 Triangle read by rows: T(n,k) is the number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2 or 3 (n >= 0, 0 <= k <= 3n).

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 1, 3, 7, 10, 10, 0, 0, 0, 1, 6, 25, 75, 175, 280, 280, 0, 0, 0, 0, 1, 10, 65, 315, 1225, 3780, 9100, 15400, 15400, 0, 0, 0, 0, 0, 1, 15, 140, 980, 5565, 26145, 102025, 323400, 800800, 1401400, 1401400, 0, 0, 0, 0, 0, 0, 1, 21, 266, 2520, 19425, 125895, 695695, 3273270, 12962950, 42042000, 106506400, 190590400, 190590400

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008, Dec 17 2008

Row n has 3n+1 entries.

			Triangle begins:
[1]
[0, 1, 1, 1]
[0, 0, 1, 3, 7, 10, 10]
[0, 0, 0, 1, 6, 25, 75, 175, 280, 280]
[0, 0, 0, 0, 1, 10, 65, 315, 1225, 3780, 9100, 15400, 15400]
[0, 0, 0, 0, 0, 1, 15, 140, 980, 5565, 26145, 102025, 323400, 800800, 1401400, 1401400]

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

See A144399, A144402, A144417, A111246 for other versions of this triangle.
Column sums give A001680, row sums give A144416. Taking last nonzero entry in each row gives A025035.
Diagonals include A000217, A001296, A027778, A144516; also A025035.
A generalization of the triangle in A144331 (and in several other entries).
Cf. A144643.

T := proc(n, k)
option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else T(n - 1, k - 1) + (k - 1)*T(n - 1, k - 2) + 1/2*(k - 1)*(k - 2)*T(n - 1, k - 3);
end if;
end proc;
for n from 0 to 12 do lprint([seq(T(n,k),k=0..3*n)]); od:

t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 3*n := t[n, k] = t[n-1, k-1] + (k-1)*t[n-1, k-2] + (1/2)*(k-1)*(k-2)*t[n-1, k-3]; t[, ] = 0; Table[t[n, k], {n, 0, 12}, {k, 0, 3*n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

T(n, k) = T(n - 1, k - 1) + (k - 1)*T(n - 1, k - 2) + (1/2)*(k - 1)*(k - 2)*T(n - 1, k - 3).

E.g.f.: Sum_{ n >= 0, k >= 0 } T(n, k) y^n x^k / k! = exp( y*(x+x^2/2+x^3/6) ). That is, the coefficient of y^n is the e.g.f. for row n. E.g. the e.g.f. for row 2 is (1/2)*(x+x^2/2+x^3/6)^2 = 1*x^2/2! + 3*x^3/3! + 7*x^4/4! + 10*x^5/5! + 10*x^6/6!.

A189886 a(n) is the number of compositions of the set {1, 2, ..., n} into blocks, each of size 1, 2 or 3 (n >= 0).

Original entry on oeis.org

1, 1, 3, 13, 74, 530, 4550, 45570, 521640, 6717480, 96117000, 1512819000, 25975395600, 483169486800, 9678799930800, 207733600074000, 4755768505488000, 115681418156304000, 2979408725813520000, 80998627977002736000, 2317937034142810080000, 69649003197501567840000, 2192459412316607834400000, 72152830779716793506400000, 2477756318984329979756160000

Offset: 0

Adi Dani, Apr 29 2011

nonn

Sequences of sets, each set having no more than 3 elements.

			a(3) = 13 because all compositions of set {a,b,c} into blocks of size 1, 2, or 3 are:
1: ({a,b,c}),
2: ({a},{b,c}),
3: ({b,c},{a}),
4: ({b},{a,c}),
5: ({a,c},{b}),
6: ({c},{a,b}),
7: ({a,b},{c}),
8: ({a},{b},{c}),
9: ({a},{c},{b}),
10: ({b},{a},{c}),
11: ({b},{c},{a}),
12: ({c},{a},{b}),
13: ({c},{b},{a}).

Alois P. Heinz, Table of n, a(n) for n = 0..424
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)
Adi Dani, Compositions and partitions of sets

Cf. A144422, A000670, A001680.

Column k=3 of A276921.

A189886 := proc(n) local m, j; add(add(2^(2*m-n-j)*3^(j-m)*n!
*binomial(m,j)*binomial(j,2*j-(3*m-n)),j=0..3*m-n),m=0..n) end:
seq(A189886(n),n=0..24); # Peter Luschny, May 02 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n, i), i=1..min(n, 3)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 22 2016
# third Maple program:
a:= n-> n! * (<<0|1|0>, <0|0|1>, <1/6|1/2|1>>^n)[3, 3]:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 22 2016

Table[Sum[n!/(2^(n+j-2m)3^(m-j))*Binomial[m,j]*Binomial[j,n+2j-3m], {m,0,n},{j,0,3m-n}],{n,0,15}]

a(n)=sum(m=0,n, sum(j=0,3*m-n, n!/(2^(n+j-2*m) *3^(m-j)) *binomial(m,j) *binomial(j,n+2*j-3*m))); /* Joerg Arndt, May 03 2011 */

a(n) = sum(m=0..n, sum(j=0..3*m-n, n!/(2^(n+j-2*m) * 3^(m-j)) * C(m,j) * C(j,n+2*j-3*m))) where C(n,k) is the binomial coefficient.
a(n) = n * a(n-1) + n*(n-1)/2 * a(n-2) + n*(n-1)*(n-2)/6 * a(n-3). - Istvan Mezo, Jun 06 2013
E.g.f.: 1/(1 - x - x^2/2 - x^3/6). -  Geoffrey Critzer, Dec 04 2012

A001681 The partition function G(n,4).

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 196, 827, 3795, 18755, 99146, 556711, 3305017, 20655285, 135399720, 927973061, 6631556521, 49294051497, 380306658250, 3039453750685, 25120541332271, 214363100120051, 1885987611214092, 17085579637664715, 159185637725413675

Offset: 0

N. J. A. Sloane

Number of '12-3 and 321-4'-avoiding permutations.
Set partitions into sets of size at most 4. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]
Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013

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).

Seiichi Manyama, Table of n, a(n) for n = 0..609 (terms 0..200 from Alois P. Heinz)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv preprint arXiv:1210.6908 [math.CO], 2012.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 19
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
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.

Cf. A001680, A276924.

Column k=4 of A229223.

G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
a:= n-> G(n, 4):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 20 2012
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n-1, i-1), i=1..min(n, 4)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2016
# Recurrence:
rec := {(-n^3-6*n^2-11*n-6)*f(n) + (-3*n^2-15*n-18)*f(n+1) + (-6*n-18)*f(n+2) - 6*f(n+3) + 6*f(n+4)=0, f(0)=1, f(1)=1, f(2)=2, f(3)=5}:
aList := gfun:-rectoproc(rec, f(n), list): aList(24); # Peter Luschny, Feb 26 2018

g[n_, k_] := g[n, k] = If[n == 0, 1, If[k<1, 0, Sum[g[n-k*j, k-1]*n!/k!^j/(n-k*j)!/j!, {j, 0, n/k}]]]; Table[g[n, 4], {n, 0, 24}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)

A001681(n)=n!*sum(k=1,n, 1/k!*sum(j=0,k, binomial(k,j)*sum(i=j,n-k+j, binomial(j,i-j)*binomial(k-j,n-3*k+3*j-i)*2^(5*k-4*j+i-2*n)*3^(j-k))));
vector(33,n,A001681(n-1)) /* Joerg Arndt, Jan 25 2011 */

x='x+O('x^66); Vec(serlaplace(exp(sum(j=1,4,x^j/j!)))) \\ Joerg Arndt, Mar 11 2014

E.g.f.: exp( x + x^2/2 + x^3/6 + x^4/24 ). - Ralf Stephan, Apr 22 2004
a(n) = n! * sum(k=1..n, 1/k! * sum(j=0..k, C(k,j) * sum(i=j..n-k+j, C(j,i-j) * C(k-j,n-3*k+3*j-i) * 2^(5*k-4*j+i-2*n) * 3^(j-k)))). [Vladimir Kruchinin, Jan 25 2011]
a(n) = G(n,4) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012
Recurrence: 6*a(n) = 6*a(n-1) + 6*(n-1)*a(n-2) + 3*(n-2)*(n-1)*a(n-3) + (n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Sep 15 2013
a(n) ~ n^(3*n/4)*exp(31*(6*n)^(1/4)/64 + 5*sqrt(6*n)/16 + (6*n)^(3/4)/6 - 3*n/4 - 21/32)/(2*6^(n/4)) * (1 + 1599*6^(3/4)/(40960*n^(1/4)) + 280873603/1677721600*sqrt(6/n) + 33870741297579 /240518168576000 *6^(1/4)/n^(3/4)). - Vaclav Kotesovec, Sep 15 2013

More terms from Ralf Stephan, Apr 22 2004

A110038 The partition function G(n,5).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 869, 4075, 20645, 112124, 648649, 3976633, 25719630, 174839120, 1245131903, 9263053753, 71806323461, 578719497070, 4839515883625, 41916097982471, 375401824277096, 3471395994487422, 33099042344383885, 325005134436155395

Offset: 0

N. J. A. Sloane, May 13 2009

nonn

Set partitions into sets of size at most 5. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]

Seiichi Manyama, Table of n, a(n) for n = 0..591 (terms 0..200 from Alois P. Heinz)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.

The sequences G(n,1), G(n,2), G(n,3), G(n,4), G(n,5), G(n,6) are given by A000012, A000085, A001680, A001681, A110038, A148092 respectively.
Column k=5 of A229223.
Cf. A276925.

G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
a:= n-> G(n, 5):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 20 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [1, 1, 2, 5, 15][n+1],
      a(n-1)+(n-1)*(a(n-2)+(n-2)/2*(a(n-3)+(n-3)/3*(a(n-4)
      +(n-4)/4*a(n-5)))))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2013

G[n_, i_] := G[n, i] = If[n == 0, 1, If[i<1, 0, Sum[G[n-i*j, i-1] *n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; a[n_] := G[n, 5]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

E.g.f.: exp( x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ).
a(n) = n! * sum(k=1..n, 1/k! * sum(r=0..k, C(k,r) * sum(m=0..r, 2^(m-r) * C(r,m) * sum(j=0..m, C(m,j) * C(j,n-m-k-j-r) * 6^(j-m) * 24^(n-r-m-k-2*j) * 120^(m+k+j+r-n))))). - Vladimir Kruchinin, Jan 25 2011
a(n) = G(n,5) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012

A323950 Number of ways to split an n-cycle into connected subgraphs, none having exactly two vertices.

Original entry on oeis.org

1, 1, 1, 2, 6, 12, 23, 44, 82, 149, 267, 475, 841, 1484, 2613, 4595, 8074, 14180, 24896, 43702, 76705, 134622, 236260, 414623, 727629, 1276917, 2240851, 3932438, 6900967, 12110373, 21252244, 37295110, 65448378, 114853920, 201554603, 353703696, 620706742

Offset: 0

Gus Wiseman, Feb 10 2019

			The a(1) = 1 through a(5) = 12 partitions:
  {{1}}  {{1}{2}}  {{123}}      {{1234}}        {{12345}}
                   {{1}{2}{3}}  {{1}{234}}      {{1}{2345}}
                                {{123}{4}}      {{1234}{5}}
                                {{124}{3}}      {{1235}{4}}
                                {{134}{2}}      {{1245}{3}}
                                {{1}{2}{3}{4}}  {{1345}{2}}
                                                {{1}{2}{345}}
                                                {{1}{234}{5}}
                                                {{123}{4}{5}}
                                                {{125}{3}{4}}
                                                {{145}{2}{3}}
                                                {{1}{2}{3}{4}{5}}

Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-3,1).

Cf. A000325, A001610, A001680, A005251, A066982, A306351, A323950, A323951, A323952, A323954.

cyceds[n_,k_]:=Union[Sort/@Join@@Table[1+Mod[Range[i,j]-1,n],{i,n},{j,Prepend[Range[i+k,n+i-1],i]}]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[cyceds[n,2],Range[n]]],{n,15}]

G.f.: (x^7-3*x^6+3*x^5-2*x^4+x^3-3*x^2+3*x-1)/((x^3-x^2+2*x-1)*(x-1)^2). - Alois P. Heinz, Feb 10 2019

More terms from Alois P. Heinz, Feb 10 2019

A323954 Regular triangle read by rows where T(n, k) is the number of ways to split an n-cycle into connected subsequences of sizes > k, n >=1, 0 <= k < n.

Original entry on oeis.org

1, 2, 1, 5, 1, 1, 12, 3, 1, 1, 27, 6, 1, 1, 1, 58, 12, 4, 1, 1, 1, 121, 22, 8, 1, 1, 1, 1, 248, 39, 13, 5, 1, 1, 1, 1, 503, 67, 22, 10, 1, 1, 1, 1, 1, 1014, 113, 36, 16, 6, 1, 1, 1, 1, 1, 2037, 188, 56, 23, 12, 1, 1, 1, 1, 1, 1, 4084, 310, 86, 35, 19, 7, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 10 2019

			Triangle begins:
     1
     2    1
     5    1    1
    12    3    1    1
    27    6    1    1    1
    58   12    4    1    1    1
   121   22    8    1    1    1    1
   248   39   13    5    1    1    1    1
   503   67   22   10    1    1    1    1    1
  1014  113   36   16    6    1    1    1    1    1
  2037  188   56   23   12    1    1    1    1    1    1
  4084  310   86   35   19    7    1    1    1    1    1    1
Row 4 counts the following partitions:
  {{1234}}        {{1234}}    {{1234}}  {{1234}}
  {{1}{234}}      {{12}{34}}
  {{12}{34}}      {{14}{23}}
  {{123}{4}}
  {{124}{3}}
  {{134}{2}}
  {{14}{23}}
  {{1}{2}{34}}
  {{1}{23}{4}}
  {{12}{3}{4}}
  {{14}{2}{3}}
  {{1}{2}{3}{4}}

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Column k = 0 is A000325. Column k = 1 is A066982. Column k = 2 is A323951. Column k = 3 is A306351.

Cf. A001610, A001680, A005251, A323950, A323951, A323952, A323953.

cycedsprop[n_,k_]:=Union[Sort/@Join@@Table[1+Mod[Range[i,j]-1,n],{i,n},{j,i+k,n+i-1}]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[cycedsprop[n,k],Range[n]]],{n,12},{k,0,n-1}]

T(n,k) = 1 - n + sum(i=1, n\(k+1), n*binomial(n-i*k-1, i-1)/i) \\ Andrew Howroyd, Jan 19 2023

T(n,k) = 1 - n + Sum_{i=1..floor(n/(k+1))} n*binomial(n-i*k-1, i-1)/i. - Andrew Howroyd, Jan 19 2023

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

A080510 Triangle read by rows: T(n,k) gives the number of set partitions of {1,...,n} with maximum block length k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A229223 Number G(n,k) of set partitions of {1,...,n} into sets of size at most k; triangle G(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A188416 T(n,k)=Number of (n*k)Xk binary arrays with rows in nonincreasing order, n ones in every column and no more than 3 ones in any row.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A144385 Triangle read by rows: T(n,k) is the number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2 or 3 (n >= 0, 0 <= k <= 3n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A189886 a(n) is the number of compositions of the set {1, 2, ..., n} into blocks, each of size 1, 2 or 3 (n >= 0).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A001681 The partition function G(n,4).