cp's OEIS Frontend

Also dimensions of free Lie algebras - see A059966, which is essentially the same sequence.

This sequence also represents the number N of cycles of length L in a digraph under x^2 seen modulo a Mersenne prime M_q=2^q-1. This number does not depend on q and L is any divisor of q-1. See Theorem 5 and Corollary 3 of the Shallit and Vasiga paper: N=sum(eulerphi(d)/order(d,2)) where d is a divisor of 2^(q-1)-1 such that order(d,2)=L. - Tony Reix, Nov 17 2005

Except for a(0) = 1, Bau-Sen Du's [1985/2007] Table 1, p. 6, has this sequence as the 7th (rightmost) column. Other columns of the table include (but are not identified as) A006206-A006208. - Jonathan Vos Post, Jun 18 2007

"Number of binary Lyndon words" means: number of binary strings inequivalent modulo rotation (cyclic permutation) of the digits and not having a period smaller than n. This provides a link to A103314, since these strings correspond to the inequivalent zero-sum subsets of U_m (m-th roots of unity) obtained by taking the union of U_n (n|m) with 0 or more U_d (n | d, d | m) multiplied by some power of exp(i 2Pi/n) to make them mutually disjoint. (But not all zero-sum subsets of U_m are of that form.) - M. F. Hasler, Jan 14 2007

Also the number of dynamical cycles of period n of a threshold Boolean automata network which is a quasi-minimal positive circuit of size a multiple of n and which is updated in parallel. - Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Feb 25 2009

Also, the number of periodic points with (minimal) period n in the iteration of the tent map f(x):=2min{x,1-x} on the unit interval. - Pietro Majer, Sep 22 2009

Number of distinct cycles of minimal period n in a shift dynamical system associated with a totally disconnected hyperbolic iterated function system (see Barnsley link). - Michel Marcus, Oct 06 2013

From Jean-Christophe Hervé, Oct 26 2014: (Start)

For n > 0, a(n) is also the number of orbits of size n of the transform associated to the Kolakoski sequence A000002 (and this is true for any map with 2^n periodic points of period n). The Kolakoski transform changes a sequence of 1's and 2's by the sequence of the lengths of its runs. The Kolakoski sequence is one of the two fixed points of this transform, the other being the same sequence without the initial term. A025142 and A025143 are the periodic points of the orbit of size 2. A027375(n) = n*a(n) gives the number of periodic points of minimal period n.

For n > 1, this sequence is equal to A059966 and to A060477, and for n = 1, a(1) = A059966(1)+1 = A060477(1)-1; this because the n-th term of all 3 sequences is equal to (1/n)*sum_{d|n} mu(n/d)*(2^d+e), with e = -1/0/1 for resp. A059966/this sequence/A060477, and sum_{d|n} mu(n/d) equals 1 for n = 1 and 0 for all n > 1. (End)

Warning: A000031 and A001037 are easily confused, since they have similar formulas.

From Petros Hadjicostas, Jul 14 2020: (Start)

Following Kam Cheong Au (2020), let d(w,N) be the dimension of the Q-span of weight w and level N of colored multiple zeta values (CMZV). Here Q are the rational numbers.

Deligne's bound says that d(w,N) <= D(w,N), where 1 + Sum_{w >= 1} D(w,N)*t^w = (1 - a*t + b*t^2)^(-1) when N >= 3, where a = phi(N)/2 + omega(N) and b = omega(N) - 1 (with omega(N) = A001221(N) being the number of distinct primes of N).

For N = 3, a = phi(3)/2 + omega(3) = 2/2 + 1 = 2 and b = omega(3) - 1 = 0. It follows that D(w, N=3) = A000079(w) = 2^w.

For some reason, Kam Cheong Au (2020) assumes Deligne's bound is tight, i.e., d(w,N) = D(w,N). He sets Sum_{w >= 1} c(w,N)*t^w = log(1 + Sum_{w >= 1} d(w,N)*t^w) = log(1 + Sum_{w >= 1} D(w,N)*t^w) = -log(1 - a*t + b*t^2) for N >= 3.

For N = 3, we get that c(w, N=3) = A000079(w)/w = 2^w/w.

He defines d*(w,N) = Sum_{k | w} (mu(k)/k)*c(w/k,N) to be the "number of primitive constants of weight w and level N". (Using the terminology of A113788, we may perhaps call d*(w,N) the number of irreducible colored multiple zeta values at weight w and level N.)

Using standard techniques of the theory of g.f.'s, we can prove that Sum_{w >= 1} d*(w,N)*t^w = Sum_{s >= 1} (mu(s)/s) Sum_{k >= 1} c(k,N)*(t^s)^k = -Sum_{s >= 1} (mu(s)/s)*log(1 - a*t^s + b*t^(2*s)).

For N = 3, we saw that a = 2 and b = 0, and hence d*(w, N=3) = a(w) = Sum_{k | w} (mu(k)/k) * 2^(w/k) / (w/k) = (1/w) * Sum_{k | w} mu(k) * 2^(w/k) for w >= 1. See Table 1 on p. 6 in Kam Cheong Au (2020). (End)

Number of ordered trees with n edges and height at most 4.

Number of palindromic structures using a maximum of three different symbols. - Marks R. Nester

Number of compositions of all even natural numbers into n parts <= 2 (0 is counted as a part), see example. - Adi Dani, May 14 2011

Consider the mapping f(a/b) = (a + 2*b)/(2*a + b). Taking a = 1, b = 2 to start with, and carrying out this mapping repeatedly on each new (reduced) rational number gives the sequence 1/2, 4/5, 13/14, 40/41, ... converging to 1. The sequence contains the denominators = (3^n+1)/2. The same mapping for N, i.e., f(a/b) = (a + N*b)/(a+b) gives fractions converging to N^(1/2). - Amarnath Murthy, Mar 22 2003

Second binomial transform of the expansion of cosh(x). - Paul Barry, Apr 05 2003

The sequence (1, 1, 2, 5, ...) = 3^n/6 + 1/2 + 0^n/3 has binomial transform A007581. - Paul Barry, Jul 20 2003

Number of (s(0), s(1), ..., s(2n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1, 2, ..., 2n+2, s(0) = 1, s(2n+2) = 1. - Herbert Kociemba, Jun 10 2004

Density of regular language L over {1,2,3}^* (i.e., number of strings of length n in L) described by regular expression 11*+11*2(1+2)*+11*2(1+2)*3(1+2+3)*. - Nelma Moreira, Oct 10 2004

Sums of rows of the triangle in A119258. - Reinhard Zumkeller, May 11 2006

Number of n-words from the alphabet A = {a,b,c} which contain an even number of a's. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 30 2006

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x = y. - Ross La Haye, Jan 10 2008

a(n+1) gives the number of primitive periodic multiplex juggling sequences of length n with base state <2>. - Steve Butler, Jan 21 2008

a(n) is also the number of idempotent order-preserving and order-decreasing partial transformations (of an n-chain). - Abdullahi Umar, Oct 02 2008

Equals row sums of triangle A147292. - Gary W. Adamson, Nov 05 2008

Equals leftmost column of A071919^3. - Gary W. Adamson, Apr 13 2009

A010888(a(n))=5 for n >= 2, that is, the digital root of the terms >= 5 equals 5. - Parthasarathy Nambi, Jun 03 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,2). - Milan Janjic, Jan 27 2010

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=6, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^(n-1)*charpoly(A,3). - Milan Janjic, Feb 21 2010

It appears that if s(n) is a rational sequence of the form s(1)=2, s(n)= (2*s(n-1)+1)/(s(n-1)+2), n>1 then s(n)=a(n)/(a(n-1)-1).

Form an array with m(1,n)=1 and m(i,j) = Sum_{k=1..i-1} m(k,j) + Sum_{k=1..j-1} m(i,k), which is the sum of the terms to the left of m(i,j) plus the sum above m(i,j). The sum of the terms in antidiagonal(n-1) = a(n). - J. M. Bergot, Jul 16 2013

From Peter Bala, Oct 29 2013: (Start)

An Engel expansion of 3 to the base b := 3/2 as defined in A181565, with the associated series expansion 3 = b + b^2/2 + b^3/(2*5) + b^4/(2*5*14) + .... Cf. A034472.

More generally, for a positive integer n >= 3, the sequence [1, n - 1, n^2 - n - 1, ..., ( (n - 2)*n^k + 1 )/(n - 1), ...] is an Engel expansion of n/(n - 2) to the base n/(n - 1). Cases include A007583 (n = 4), A083065 (n = 5) and A083066 (n = 6). (End)

Diagonal elements (and one more than antidiagonal elements) of the matrix A^n where A=(2,1;1,2). - David Neil McGrath, Aug 17 2014

From M. Sinan Kul, Sep 07 2016: (Start)

a(n) is equal to the number of integer solutions to the following equation when x is equal to the product of n distinct primes: 1/x = 1/y + 1/z where 0 < x < y <= z.

If z = k*y where k is a fraction >= 1 then the solutions can be given as: y = ((k+1)/k)*x and z = (k+1)*x.

Here k can be equal to any divisor of x or to the ratio of two divisors.

For example for x = 2*3*5 = 30 (product of three distinct primes), k would have the following 14 values: 1, 6/5, 3/2, 5/3, 2, 5/2, 3, 10/3, 5, 6, 15/2, 10, 15, 30.

As an example for k = 10/3, we would have y=39, z=130 and 1/39 + 1/130 = 1/30.

Here finding the number of fractions would be equivalent to distributing n balls (distinct primes) to two bins (numerator and denominator) with no empty bins which can be found using Stirling numbers of the second kind. So another definition for a(n) is: a(n) = 2^n + Sum_{i=2..n} Stirling2(i,2)*binomial(n,i).

(End)

a(n+1) is the smallest i for which the Catalan number C(i) (see A000108) is divisible by 3^n for n > 0. This follows from the rule given by Franklin T. Adams-Watters for determining the multiplicity with which a prime divides C(n). We need to find the smallest number in base 3 to achieve a given count. Applied to prime 3, 1 is the smallest digit that counts but requires to be followed by 2 which cannot be at the end to count. Therefore the number in base 3 of the form 1{n-1 times}20 = (3^(n+1) + 1)/2 + 1 = a(n+1)+1 is the smallest number to achieve count n which implies the claim. - Peter Schorn, Mar 06 2020

Let A be a Toeplitz matrix of order n, defined by: A[i,j]=1, if ij; A[i,i]=2. Then, for n>=1, a(n) = det A. - Dmitry Efimov, Oct 28 2021

a(n) is the least number k such that A065363(k) = -(n-1), for n > 0. - Amiram Eldar, Sep 03 2022

All step shifts of a sequence are considered to be equivalent, where a step shift transformation is obtained by selecting every k-th element of a sequence for some k relatively prime to n. For example, 2 is relatively prime to 5 and a 2-step shift of abcde is bdace.

Equivalently, walks on triangle, visiting n+2 vertices, so length n+1, n "corners"; the symmetry group is S3, reversing a walk does not count as different. Walks are not self-avoiding. - Colin Mallows

Slavik V. Jablan observes that this is also the number of rational knots and links with n+2 crossings (cf. A018240). See reference. [Corrected by Andrey Zabolotskiy, Jun 18 2020]

Number of bit strings of length (n-1), not counting strings which are the end-for-end reversal or the 0-for-1 reversal of each other as different. - Carl Witty (cwitty(AT)newtonlabs.com), Oct 27 2001

The formula given in page 1095 of the Balasubramanian reference can be used to derive this sequence. - Parthasarathy Nambi, May 14 2007

Also number of compositions of n up to direction, where a composition is considered equivalent to its reversal, see example. - Franklin T. Adams-Watters, Oct 24 2009

Number of normally non-isomorphic realizations of the associahedron of type I starting with dimension 2 in Ceballos et al. - Tom Copeland, Oct 19 2011

Number of fibonacenes with n+2 hexagons. See the Balaban and the Dobrynin references. - Emeric Deutsch, Apr 21 2013

From the point of view of binary grids, it is a (1,n)-rectangular grid. A225826 to A225834 are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11. - Yosu Yurramendi, May 19 2013

Number of n-vertex difference graphs (bipartite 2K_2-free graphs) [Peled & Sun, Thm. 9]. - Falk Hüffner, Jan 10 2016

The offset should be 0, since the first row of A034851 is row 0. The name would then be: "Number of n bead...". - Daniel Forgues, Jul 26 2018

a(n) is the number of non-isomorphic generalized rigid ladders with n cells. A generalized rigid ladder with n cells is a graph with vertex set is the union of {u_0, u_1, ..., u_n} and {v_0, v_1, ..., v_n}, and for every 0 <= i <= n-1, the edges are of the form {u_i,u_i+1}, {v_i, v_i+1}, {u_i,v_i} and either {u_i,v_i+1} or {u_i+1,v_i}. - Christian Barrientos, Jul 29 2018

Also number of non-isomorphic stairs with n+1 cells. A stair is a snake polyomino allowing only two directions for adjacent cells: east and north. - Christian Barrientos and Sarah Minion, Jul 29 2018

From Robert A. Russell, Oct 28 2018: (Start)

There are two different unoriented row colorings using two colors that give us very similar results here, a difference of one in the offset. In an unoriented row, chiral pairs are counted as one.

a(n) is the number of color patterns (set partitions) of an unoriented row of length n using two or fewer colors (subsets). Two color patterns are equivalent if the colors are permutable.

a(n+1) is the number of ways to color an unoriented row of length n using two noninterchangeable colors (one need not use both colors).

See the examples below of these two different colorings. (End)

Also arises from the enumeration of types of based polyhedra with exactly two triangular faces [Rademacher]. - N. J. A. Sloane, Apr 24 2020

a(n) is the number of (unlabeled) 2-paths with n+4 vertices. (A 2-path with order n at least 4 can be constructed from a 3-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to an existing 2-clique containing an existing 2-leaf.) - Allan Bickle, Apr 05 2022

a(n) is the number of caterpillars with a perfect matching and order 2n+2. - Christian Barrientos, Sep 12 2023

a(n) is also the number of distinct planar embeddings of the (n+2)-centipede graph (up to at least n=8 and likely for all larger n). - Eric W. Weisstein, May 21 2024

a(n) is also the number of distinct planar embeddings of the 2 X (n+2) grid graph i.e., the (n+2)-ladder graph. - Eric W. Weisstein, May 21 2024

Dimension of the homogeneous component of degree n of the free Jordan algebra on two generators (or, in this case, the free special Jordan algebra on two generators). It follows from (Shirshov 1956, Cohn 1959). - Vladimir Dotsenko, Mar 29 2025

Number of palindromic structures using exactly three different symbols; Mobius transform: A056279. - Marks R. Nester

Number of ways of placing n labeled balls into k=3 indistinguishable boxes. - Thomas Wieder, Nov 30 2004

With two leading zeros, this is the second binomial transform of cosh(x)-1 and the binomial transform of A000225 (with extra leading zero). - Paul Barry, May 13 2003

Let [m] denote the first m positive integers. Then a(n) is the number of functions f from [n] to [n+1] that satisfy (i) f(x) > x for all x, (ii) f(x) = n+1 for exactly 3 elements and (iii) f(f(x)) = n+1 for the remaining n-3 elements of [n]. For example, a(4)=6 since there are exactly 6 functions from {1,2,3,4} to {1,2,3,4,5} such that f(x) > x, f(x) = 5 for 3 elements and f(f(x)) = 5 for the remaining element. The functions are f1 = {(1,5), (2,5), (3,4), (4,5)}, f2 = {(1,5), (2,3), (3,5), (4,5)}, f3 = {(1,5), (2,4), (3,5), (4,5)}, f4 = {(1,2), (2,5), (3,5), (4,5)}, f5 = {(1,3), (2,5), (3,5), (4,5)}, f6 = {(1,4), (2,5), (3,5), (4,5)}. - Dennis P. Walsh, Feb 20 2007

Conjecture. Let S(1)={1} and, for n > 1, let S(n) be the smallest set containing x, 2x and 3x for each element x in S(n-1). Then a(n+2) is the sum of the elements in S(n). (It is easy to prove that the number of elements in S(n) is the n-th triangular number given by A001952.) See A122554 for a sequence defined in this way. - John W. Layman, Nov 21 2007; corrected (a(n) to a(n+2) due to offset change) by Fred Daniel Kline, Oct 02 2014

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which x and y are disjoint and for which x is not a subset of y and y is not a subset of x. Wieder calls these "disjoint strict 2-combinations". - Ross La Haye, Jan 11 2008; corrected by Ross La Haye, Oct 29 2008

Also, let P(A) be the power set of an n-element set A. Then a(n+2) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 11 2008

3 * a(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k+1) for k = 0, 1, ..., n. - Michael Somos, Apr 29 2012

John W. Layman's conjecture that a(n+2) is the sum of elements in S(n) follows from the identification of S(n) with the first n rows of A036561, whose row sums are A001047. - Fred Daniel Kline, Oct 02 2014

From M. Sinan Kul, Sep 08 2016: (Start)

Let m be equal to the product of n-1 distinct primes. Then a(n) is equal to the number of distinct fractions >=1 that may be created by dividing a divisor of m by another divisor. For example for m = 2*3*5 = 30, we would have the following 6 fractions: 6/5, 3/2, 5/3, 5/2, 10/3, 15/2.

Here finding the number of fractions would be equivalent to distributing n-1 balls (distinct primes) to two bins (numerator and denominator) with no empty bins which can be found by Stirling numbers of the second kind. So another definition for a(n) is a(n) = Sum_{i=2..n-1} Stirling2(i,2)*binomial(n-1,i).

Also for n > 0, a(n) = (d(m^2)+1)/2 - d(m) where m is equal to the product of n-1 distinct primes. Example for a(5): m = 2*3*5*7 = 210 (product of four distinct primes) so a(5) = (d(210^2)+1)/2 - d(210) = 41 - 16 = 25. (End)

6*a(n) is the number of ternary strings of length n that contain at least one of each of the 3 symbols on which they are defined. For example, for n=4, the strings are the 12 permutations of 0012, the 12 permutations of 0112, and the 12 permutations of 0122. - Enrique Navarrete, Aug 23 2021

A simpler form of La Haye's first comment is: a(n+1) is the number of ways we can form disjoint unions of two nonempty subsets of [n] (see example below). Cf. A001047 for the requirement that the union contains n. - Enrique Navarrete, Aug 24 2021

As partial sums of the Nicomachus triangle's rows and the differences of the powers of 3 and 2 (A001047), each iteration corresponds to two figurate variations of the Sierpinski triangle (3^n) with cross-correlation to the Nicomachus triangle, see illustrations in links. The Sierpinski half-hexagons of (A001047) stack and conform to the footprint of 2^n - 1 triangular numbers. The 3^n Sierpinski triangle minus its 2^n bottom row, also correlates to the Nicomachus triangle according to each Sierpinski triangular sub-row. - John Elias, Oct 04 2021

Number of palindromic structures using a maximum of four different symbols. - Marks R. Nester

Dimension of the universal embedding of the symplectic dual polar space DSp(2n,2) (conjectured by A. Brouwer, proved by P. Li). - J. Taylor (jt_cpp(AT)yahoo.com), Apr 02 2004.

Apart from initial term, same as A124303. - Valery A. Liskovets, Nov 16 2006

Hankel transform is := [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008

a(n) is also the number of distinct solutions (avoiding permutations) to the equation: XOR(A,B,C)=0 where A,B,C are n-bit binary numbers. - Ramasamy Chandramouli, Jan 11 2009

The rank of the fundamental group of the Z_p^n - cobordism category in dimension 1+1 for the case p=2 (see paper below). The expression for any prime p is (p^(2n-1)+p^(n+1)-p^(n-1)+p^2-p-1)/(p^2-1). - Carlos Segovia Gonzalez, Dec 05 2012

The number of isomorphic classes of regular four coverings of a graph with respect to the identity automorphism (S. Hong and J. H. Kwak). - Carlos Segovia Gonzalez, Aug 01 2013

The density of a language with four letters (N. Moreira and R. Reis). - Carlos Segovia Gonzalez, Aug 01 2013

The wire stays in the plane, there are n bends, each is R,L or O; turning the wire over does not count as a new figure.

Equivalently, walks of n+1 steps on a tetrahedron, visiting n+2 vertices, with n "corners"; the symmetry group is S4, reversing a walk does not count as different. Simply interpret R,L,O as instructions to turn R, turn L, or retrace the last step. Walks are not self-avoiding.

Also, it appears that a(n) gives the number of equivalence classes of n-tuples of 0, 1 and 2, where two n-tuples are equivalent if one can be obtained from the other by a sequence of operations R and C, where R denotes reversal and C denotes taking the 2's complement (C(x)=2-x). This has been verified up to a(19)=290585050. Example: for n=3 there are ten equivalence classes {000, 222}, {001, 100, 122, 221}, {002, 022, 200, 220}, {010, 212}, {011, 110, 112, 211}, {012, 210}, {020, 202}, {021, 102, 120, 201}, {101, 121}, {111}, so a(3)=10. - John W. Layman, Oct 13 2009

There exists a bijection between chains of n+2 hexagons and the above described equivalence classes of n-tuples of 0,1, and 2. Namely, for a given chain of n+2 hexagons we take the sequence of the numbers of vertices of degree 2 (0, 1, or 2) between the consecutive contact vertices on one side of the chain; switching to the other side we obtain the 2's complement of this sequence; reversing the order of the hexagons, we obtain the reverse sequence. The inverse mapping is straightforward. For example, to a linear chain of 7 hexagons there corresponds the 5-tuple 11111. - Emeric Deutsch, Apr 22 2013

If we treat two wire bends (or walks, or tuples) related by turning over (or reversing) as different in any of the above-given interpretations of this sequence, we get A007051 (or A124302). Also, a(n-1) is the sum of first 3 terms in n-th row of A284949, see crossrefs therein. - Andrey Zabolotskiy, Sep 29 2017

a(n-1) is the number of color patterns (set partitions) in an unoriented row of length n using 3 or fewer colors (subsets). - Robert A. Russell, Oct 28 2018

From Allan Bickle, Jun 02 2022: (Start)

a(n) is the number of (unlabeled) 3-paths with n+6 vertices. (A 3-path with order n at least 5 can be constructed from a 4-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to an existing 3-clique containing an existing 3-leaf.)

Recurrences appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)

a(n) is also the number of distinct planar embeddings of the (n+1)-alkane graph (up to at least n=9, and likely for all n). - Eric W. Weisstein, May 21 2024

Number of palindromes of length n using a maximum of four different symbols.

Number of achiral rows of n colors using up to four colors. - Robert A. Russell, Nov 09 2018

Interleaving of A000302 and 4*A000302.

Unsigned version of A141125.

Binomial transform is A164907. Second binomial transform is A164908. Third binomial transform is A057651. Fourth binomial transform is A016129.

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

Density of regular language L over {1,2,3,4}^* (i.e., number of strings of length n in L) described by regular expression 11* + 11*2(1+2)* + 11*2(1+2)*3(1+2+3)* + 11*2(1+2)*3(1+2+3)*4(1+2+3+4)* + 11*2(1+2)*3(1+2+3)*4(1+2+3+4)*5(1+2+3+4+5)* - Nelma Moreira, Oct 10 2004

Number of set partitions of [n] into at most 5 parts. - Joerg Arndt, Apr 18 2014

Number of bracelet structures of length n using exactly k different colored beads. Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure. - Andrew Howroyd, Apr 06 2017

The number of achiral structures (A) is given in A140735 (odd n) and A293181 (even n). The number of achiral structures plus twice the number of chiral pairs (A+2C) is given in A152175. These can be used to determine A+C by taking half their average, as is done in the Mathematica program. - Robert A. Russell, Feb 24 2018

T(n,k)=pi_k(C_n) which is the number of non-equivalent partitions of the cycle on n vertices, with exactly k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. - Mohammad Hadi Shekarriz, Aug 21 2019