cp's OEIS Frontend

For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001

a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad Brewbaker, Jan 31 2003

a(n-1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad Brewbaker, Jan 31 2003

Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch, Aug 28 2004

Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - Simone Severini, Oct 15 2004

The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... . - Jon Perry, Sep 20 2008

Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...). - Gary W. Adamson, Dec 25 2008

First column of A092582. - Mats Granvik, Feb 08 2009

The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

For n>1: a(n) = A173333(n,2). - Reinhard Zumkeller, Feb 19 2010

Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300). - Gary W. Adamson, Aug 02 2010

For n>=2: a(n) is the number of connected 2-regular labeled graphs on (n+1) nodes (Cf. A001205). - Geoffrey Critzer, Feb 16 2011.

The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011

Also [1, 1] together with the row sums of triangle A162608. - Omar E. Pol, Mar 09 2012

a(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n-1]), in short 1123...(n-1), taken cyclically. E.g., n=2: 11, n=3: 112, n=4: 1123, 1132, 1213, n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the next-to-last entry in line n>=2 of the representative necklace partition array A212359. - Wolfdieter Lang, Jun 26 2012

For m >= 3, a(m-1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286. - Stanislav Sykora, May 10 2014

In factorial base (A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below. - Antti Karttunen, Dec 19 2015

Also (by definition) the independence number of the n-transposition graph. - Eric W. Weisstein, May 21 2017

Number of permutations of n letters containing an even number of even cycles. - Michael Somos, Jul 11 2018

Equivalent to Brewbaker's and Sykora's comments, a(n - 1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135. - Gus Wiseman, Oct 20 2018

For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaf-labeled tree topologies that have a caterpillar shape (column k=1 of A306364). - Noah A Rosenberg, Feb 11 2019

Also the clique covering number of the n-Bruhat graph. - Eric W. Weisstein, Apr 19 2019

a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020

For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number (A180332) and p_1^e_1*p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number. - Ivan N. Ianakiev, Mar 11 2020

For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cycle-mates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cycle-mates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows. - Dennis P. Walsh, May 28 2020

H_n(x,y) is defined recursively by:

H_0(x,y) = y+1;

H_1(x,0) = x;

H_2(x,0) = 0;

H_n(x,0) = 1, for n>2;

H_n(x,y) = H_{n-1}(x,H_n(x,y-1)), for integers n>0 and y>0.

Consequently:

H_0(x,y) = y+1 is the successor function on y;

H_1(x,y) = x+y is addition;

H_2(x,y) = x*y is multiplication;

H_3(x,y) = x^y is exponentiation;

H_4(x,y) = x^^y is tetration (a height-y exponential tower x^x^x^... );

...

Extending to negative-order hyperoperators via the recursive formula:

H_0(x,y) = H_{-1}(x,H_0(x,y-1)) = H_{-1}(x,y).

Therefore:

H_{-n}(x,y) = H_0(x,y), for every nonnegative n.

This function is an Ackermann function variant because it satisfies the recurrence relation above (see A046859).

Other hyperoperation notations equivalent to H_n(x,y) include:

Square Bracket or Box: a [n] b;

Conway Chain Arrows: a -> b -> n-2;

Knuth Up-arrow: a "up-arrow"(n-2) b;

Standard Caret: a ^(n-2) b.

Originally published as 3 agg-op-n 3 for n > 0. - Natan Arie Consigli, Apr 22 2015

Sequence can also be defined as a(0) = 3, a(1) = 5, a(n) = H_{n-1}(3,3) for n > 1. - Natan Arie Consigli, Apr 22 2015; edited by Danny Rorabaugh, Oct 18 2015

Before introducing the H_n notation, this sequence was named "3 agg-op-n 2, where the binary aggregation operators agg-op-n are zeration, addition, multiplication, exponentiation, superexponentiation, ..." - Danny Rorabaugh, Oct 14 2015

The next term is 3^3^...^3 (with 7625594784987 3's). - Jianing Song, Dec 25 2018

Apparently a(n) = A181131(n) for n>=2 (checked numerically up to n=640). - R. J. Mathar, Aug 25 2025

The denominators of the T(n, n+1) with T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0. For the numerators of the T(n, n+1) see A191972.

The T(n, m) are defined by A164555(n)/A027642(n) and its successive differences, see the formulas.

Reading the array T(n, m), see the examples, by its antidiagonals leads to A085737(n)/A085738(n).

A164555(n)/A027642(n) is an autosequence (eigensequence whose inverse binomial transform is the sequence signed) of the second kind; the main diagonal T(n, n) is twice the first upper diagonal T(n, n+1).

We can get the Bernoulli numbers from the T(n, n+1) in an original way, see A192456/A191302.

Also the denominators of T(n, n+1) of the table defined by A085737(n)/A085738(n), the upper diagonal, called the median Bernoulli numbers by Chen. As such, Chen proved that a(n) is even only for n=0 and n=1 and that a(n) are squarefree numbers. (see Chen link). - Michel Marcus, Feb 01 2013

The sum of the antidiagonals of T(n,m) is 1 in the first antidiagonal, otherwise 0. Paul Curtz, Feb 03 2015

The EG2[2*m,n] matrix coefficients were introduced in A008955. We discovered that EG2[2m,n] = Sum_{k = 1..n} (-1)^(k+n)*t1(n-1,k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2 with t1(n,m) the central factorial numbers A008955 and eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function.

A different way to define these matrix coefficients is EG2[2*m,n] = (1/m)*Sum_{k = 0..m-1} ZETA(2*m-2*k, n-1)*EG2[2*k, n] with ZETA(2*m, n-1) = zeta(2*m) - Sum_{k = 1..n-1} (k)^(-2*m) and EG2[0, n] = 1, for m = 0, 1, 2, ..., and n = 1, 2, 3, ... .

We define the row sums of the EG2 matrix rs(2*m,p) = Sum_{n >= 1} (n^p)*EG2(2*m,n) for p = -2, -1, 0, 1, ... and m >= p+2. We discovered that rs(2*m,p=-2) = 2*eta(2*m+2) = (1 - 2^(1-(2*m+2)))*zeta(2*m+2). This formula is quite unlike the other rs(2*m,p) formulas, see the examples.

The series expansions of the row generators RGEG2(z,2*m) about z = 0 lead to the EG2[2*m,n] coefficients while the series expansions about z = 1 lead to the ZG1[2*m-1,n] coefficients, see the formulas.

The first Maple program gives the triangle coefficients. Adding the second program to the first one gives information about the row sums rs(2*m,p).

The a(n) formulas of the right hand columns are related to sequence A036283, see also A161740 and A161741.

The next term, a(4), has about 8*10^153 decimal digits. - Charles R Greathouse IV, Nov 15 2022

Continued fraction expansion of (sqrt(5) + 1)/(2*sqrt(5)).

Inverse binomial transform is {0, 1, 4, 10, 21, 41, 78, 148, ...}, A132925 with one leading zero.

Also the main diagonal in the expansion of (1 + x)^n - 1 + x^2 (A300453).

The partial sum of this sequence is A184985.

a(n) is the number of state diagrams having n components that are obtained from an n-foil [(2,n)-torus knot] shadow. Let a shadow diagram be the regular projection of a mathematical knot into the plane, where the under/over information at every crossing is omitted. A state for the shadow diagram is a diagram obtained by merging either of the opposite areas surrounding each crossing.

a(n) satisfies the identities a(n)^a(n+k) = a(n), 2^a(k) = 2*a(k) and a(k)! = a(k), k > 0.

Also the number of non-isomorphic simple connected undirected graphs with n+1 edges and a longest path of length 2. - Nathaniel Gregg, Nov 02 2021

The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.

This sequence is periodic with a pre-period of length 3 (namely 1, 1, 2) and a period of length 1 (namely 1).

Decimal expansion of 1009/9000. - Elmo R. Oliveira, Jun 16 2024

The inverse multinomial transform [IML] transforms an input sequence b(n) into the output sequence a(n). The IML transform inverses the effect of the multinomial transform [MNL], see A274760, and is related to the logarithmic transform, see A274805 and the first formula.

To preserve the identity MNL[IML[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL.

In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.

We observe that the inverse multinomial transform leaves the value of a(0) undefined.

The Maple programs can be used to generate the inverse multinomial transform of a sequence. The first program is derived from a formula given by Alois P. Heinz for the logarithmic transform, see the first formula and A001187. The second program uses the e.g.f. for multivariate row polynomials, see A127671 and the examples. The third program uses information about the inverse of the inverse of the multinomial transform, see A274760.

The IML transform of A001818(n) = ((2*n-1)!!)^2 leads quite unexpectedly to A005411(n), a sequence related to certain Feynman diagrams.

Some IML transform pairs, n >= 1: A000110(n) and 1/A000142(n-1); A137341(n) and A205543(n); A001044(n) and A003319(n+1); A005442(n) and A000204(n); A005443(n) and A001350(n); A007559(n) and A000244(n-1); A186685(n+1) and A131040(n-1); A061711(n) and A141151(n); A000246(n) and A000035(n); A001861(n) and A141044(n-1)/A001710(n-1); A002866(n) and A000225(n); A000262(n) and A000027(n).

s(n,k), signed Stirling numbers of the first kind (A048994):

1,

0, 1,

0, -1, 1,

0, 2, -3, 1,

0, -6, 11, -6, 1

etc.

The unsigned numbers, abs(s(n,k)), are the unsigned Stirling numbers of the first kind, A132393(n).

For the integration of these triangles we divide by A002260(n+1). For the first one the reduced numbers are

1,

0, 1/2,

0, -1/2, 1/3,

0, 1, -1, 1/4,

0, -3, 11/3, -3/2, 1/5,

etc.

Hence the denominators in the example.

Sums by rows: 1, 1/2, -1/6, 1/4, -19/30, 27/12 = 9/4, = (-1)^(n+1)*A141417(n)/A002790(n) = A006232(n)/A006233(n) (*).

Because the integration is between 0 and 1, the fractions appear in a numerical Adams integration with the denominators multiplied by n!, i.e., 1, 1/2, -1/12, 1/24, -19/720, 27/1440, ... . Reference, array p. 36.

(*) The Cauchy numbers of the first type or the Bernoulli numbers of the second kind.

Without signs, the row sums are 1, 1/2, 5/6, 9/4, 251/30, 475/12, ... = A002657(n)/A002790(n), Cauchy numbers of the second type. See Nørlund numbers, 1924.

Fractions: 1/2, 1/2, -4/3, 2, -38/15, 3, -73/21, 4, -68/15, 5, -179/33, 6, -9218/1365, 7, ... .

Numerators are A307974(n).

a(2n) same as denominators of cosecant numbers A001897 for n>0 (conjectured).