cp's OEIS Frontend

The divisor in the formula is A002866(4) = 192

Apart from initial term, same as A000079 (powers of 2).

Number of compositions (ordered partitions) of n. - Toby Bartels, Aug 27 2003

Number of ways of putting n unlabeled items into (any number of) labeled boxes where every box contains at least one item. Also "unimodal permutations of n items", i.e., those which rise then fall. (E.g., for three items: ABC, ACB, BCA and CBA are unimodal.) - Henry Bottomley, Jan 17 2001

Number of permutations in S_n avoiding the patterns 213 and 312. - Tuwani Albert Tshifhumulo, Apr 20 2001. More generally (see Simion and Schmidt), the number of permutations in S_n avoiding (i) the 123 and 132 patterns; (ii) the 123 and 213 patterns; (iii) the 132 and 213 patterns; (iv) the 132 and 231 patterns; (v) the 132 and 312 patterns; (vi) the 213 and 231 patterns; (vii) the 213 and 312 patterns; (viii) the 231 and 312 patterns; (ix) the 231 and 321 patterns; (x) the 312 and 321 patterns.

a(n+2) is the number of distinct Boolean functions of n variables under action of symmetric group.

Number of unlabeled (1+2)-free posets. - Detlef Pauly, May 25 2003

Image of the central binomial coefficients A000984 under the Riordan array ((1-x), x*(1-x)). - Paul Barry, Mar 18 2005

Binomial transform of (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...); inverse binomial transform of A007051. - Philippe Deléham, Jul 04 2005

Also, number of rationals in [0, 1) whose binary expansions terminate after n bits. - Brad Chalfan, May 29 2006

Equals row sums of triangle A144157. - Gary W. Adamson, Sep 12 2008

Prepend A089067 with a 1, getting (1, 1, 3, 5, 13, 23, 51, ...) as polcoeff A(x); then (1, 1, 2, 4, 8, 16, ...) = A(x)/A(x^2). - Gary W. Adamson, Feb 18 2010

An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 2, 8, 32 and 128, lead to this sequence. For the corner squares these vectors lead to the companion sequence A094373. - Johannes W. Meijer, Aug 15 2010

From Paul Curtz, Jul 20 2011: (Start)

Array T(m,n) = 2*T(m,n-1) + T(m-1,n):

1, 1, 2, 4, 8, 16, ... = a(n)

1, 3, 8, 20, 48, 112, ... = A001792,

1, 5, 18, 56, 160, 432, ... = A001793,

1, 7, 32, 120, 400, 1232, ... = A001794,

1, 9, 50, 220, 840, 2912, ... = A006974, followed with A006975, A006976, gives nonzero coefficients of Chebyshev polynomials of first kind A039991 =

1,

1, 0,

2, 0, -1,

4, 0, -3, 0,

8, 0, -8, 0, 1.

T(m,n) third vertical: 2*n^2, n positive (A001105).

Fourth vertical appears in Janet table even rows, last vertical (A168342 array, A138509, rank 3, 13, = A166911)). (End)

A131577(n) and differences are:

0, 1, 2, 4, 8, 16,

1, 1, 2, 4, 8, 16, = a(n),

0, 1, 2, 4, 8, 16,

1, 1, 2, 4, 8, 16.

Number of 2-color necklaces of length 2n equal to their complemented reversal. For length 2n+1, the number is 0. - David W. Wilson, Jan 01 2012

Edges and also central terms of triangle A198069: a(0) = A198069(0,0) and for n > 0: a(n) = A198069(n,0) = A198069(n,2^n) = A198069(n,2^(n-1)). - Reinhard Zumkeller, May 26 2013

These could be called the composition numbers (see the second comment) since the equivalent sequence for partitions is A000041, the partition numbers. - Omar E. Pol, Aug 28 2013

Number of self conjugate integer partitions with exactly n parts for n>=1. - David Christopher, Aug 18 2014

The sequence is the INVERT transform of (1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). - Gary W. Adamson, Jul 16 2015

Number of threshold graphs on n nodes [Hougardy]. - Falk Hüffner, Dec 03 2015

Number of ternary words of length n in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017

a(n) is the number of words of length n over an alphabet of two letters, of which one letter appears an even number of times (the empty word of length 0 is included). See the analogous odd number case in A131577, and the Balakrishnan reference in A006516 (the 4-letter odd case), pp. 68-69, problems 2.66, 2.67 and 2.68. - Wolfdieter Lang, Jul 17 2017

Number of D-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are D-equivalent iff the positions of pattern D are identical in these paths. - Sergey Kirgizov, Apr 08 2018

Number of color patterns (set partitions) for an oriented row of length n using two or fewer colors (subsets). Two color patterns are equivalent if we permute the colors. For a(4)=8, the 4 achiral patterns are AAAA, AABB, ABAB, and ABBA; the 4 chiral patterns are the 2 pairs AAAB-ABBB and AABA-ABAA. - Robert A. Russell, Oct 30 2018

The determinant of the symmetric n X n matrix M defined by M(i,j) = (-1)^max(i,j) for 1 <= i,j <= n is equal to a(n) * (-1)^(n*(n+1)/2). - Bernard Schott, Dec 29 2018

For n>=1, a(n) is the number of permutations of length n whose cyclic representations can be written in such a way that when the cycle parentheses are removed what remains is 1 through n in natural order. For example, a(4)=8 since there are exactly 8 permutations of this form, namely, (1 2 3 4), (1)(2 3 4), (1 2)(3 4), (1 2 3)(4), (1)(2)(3 4), (1)(2 3)(4), (1 2)(3)(4), and (1)(2)(3)(4). Our result follows readily by conditioning on k, the number of parentheses pairs of the form ")(" in the cyclic representation. Since there are C(n-1,k) ways to insert these in the cyclic representation and since k runs from 0 to n-1, we obtain a(n) = Sum_{k=0..n-1} C(n-1,k) = 2^(n-1). - Dennis P. Walsh, May 23 2020

Maximum number of preimages that a permutation of length n + 1 can have under the consecutive-231-avoiding stack-sorting map. - Colin Defant, Aug 28 2020

a(n) is the number of occurrences of the empty set {} in the von Neumann ordinals from 0 up to n. Each ordinal k is defined as the set of all smaller ordinals: 0 = {}, 1 = {0}, 2 = {0,1}, etc. Since {} is the foundational element of all ordinals, the total number of times it appears grows as powers of 2. - Kyle Wyonch, Mar 30 2025

Another name: Triangle of signless Stirling numbers of the first kind.

Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938.

A094645*A007318 as infinite lower triangular matrices.

Row sums are the factorial numbers. - Roger L. Bagula, Apr 18 2008

Exponential Riordan array [1/(1-x), log(1/(1-x))]. - Ralf Stephan, Feb 07 2014

Also the Bell transform of the factorial numbers (A000142). For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015

This is the lower triagonal Sheffer matrix of the associated or Jabotinsky type |S1| = (1, -log(1-x)) (see the W. Lang link under A006232 for the notation and references). This implies the e.g.f.s given below. |S1| is the transition matrix from the monomial basis {x^n} to the rising factorial basis {risefac(x,n)}, n >= 0. - Wolfdieter Lang, Feb 21 2017

T(n, k), for n >= k >= 1, is also the total volume of the n-k dimensional cell (polytope) built from the n-k orthogonal vectors of pairwise different lengths chosen from the set {1, 2, ..., n-1}. See the elementary symmetric function formula for T(n, k) and an example below. - Wolfdieter Lang, May 28 2017

From Wolfdieter Lang, Jul 20 2017: (Start)

The compositional inverse w.r.t. x of y = y(t;x) = x*(1 - t(-log(1-x)/x)) = x + t*log(1-x) is x = x(t;y) = ED(y,t) := Sum_{d>=0} D(d,t)*y^(d+1)/(d+1)!, the e.g.f. of the o.g.f.s D(d,t) = Sum_{m>=0} T(d+m, m)*t^m of the diagonal sequences of the present triangle. See the P. Bala link for a proof (there d = n-1, n >= 1, is the label for the diagonals).

This inversion gives D(d,t) = P(d, t)/(1-t)^(2*d+1), with the numerator polynomials P(d, t) = Sum_{m=0..d} A288874(d, m)*t^m. See an example below. See also the P. Bala formula in A112007. (End)

For n > 0, T(n,k) is the number of permutations of the integers from 1 to n which have k visible digits when viewed from a specific end, in the sense that a higher value hides a lower one in a subsequent position. - Ian Duff, Jul 12 2019

For n >= 3, this polynomial is the Poincaré polynomial (or growth series) for the reflection group (or Weyl group, or finite Coxeter group) D_n.

Row lengths are 1, 3, 7, 13, 21, 31, 43, 57, ...: see A002061. - Michel Marcus, May 17 2013

The asymptotic growth of maximum elements for the reflection group D_n is about 2(n-1/2) (compare with A000140). - Mikhail Gaichenkov, Aug 21 2019

Row maxima ~ 2^(n-1)*n!/(sigma * sqrt(3/Pi)), sigma^2 = (4*n^3 - 3*n^2 - n)/36 = variance of D_n. - Mikhail Gaichenkov, Feb 08 2023

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:

a\b 1.......2.......3.......4.......5.......6

-1 A144431

0 A007318 A038208 A038221

1 A008292 A256890 A257180 A257606 A257607

2 A060187 A257609 A257611 A257613 A257615

3 A142458 A257610 A257620 A257622 A257624 A257626

4 A142459 A257612 A257621

5 A142460 A257614 A257623

6 A142461 A257616 A257625

7 A142462 A257617 A257627

8 A167884 A257618

9 A257608 A257619

The row sums of these, and similarly constructed number triangles, are shown in the following table:

a\b 1.......2.......3.......4.......5.......6.......7.......8.......9

0 A000079 A000302 A000400

1 A000142 A001715 A001725 A049388 A049198

2 A000165 A002866 A002866 A051580 A051582

3 A008544 A051578 A037559 A051605 A051607 A051609

4 A001813 A047053 A000407 A034177 A051618 A051620 A051622

5 A047055 A008546 A008548 A034300 A034325 A051688 A051690

6 A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103

7 A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105

8 A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115

9 A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023

10 A051262 A035265 A035273 A035277

11 A254322

12 A145448

The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).

a\b 0 1 2 3

-2 A130595/1

-1

0

1 A110555/-1 A120434/-1 A144697/1 A144699/2

With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).

If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).

In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019

Row m = 3 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

Let b(n) = LPT[ A001147 ] = -A001147(n-1) for n > 0 and 1 for n=0, where LPT represents the action of the list partition transform described in A133314.

Then T(n,k) = binomial(n,k) * b(n-k) .

Form the matrix of polynomials TB(n,k,t) = T(n,k) * t^(n-k) = binomial(n,k) * b(n-k) * t^(n-k) = binomial(n,k) * Pb(n-k,t),

beginning as

1;

-1, 1;

-1*t, -2, 1;

-3*t^2, -3*t, -3, 1;

-15*t^3, -12*t^2, -6*t, -4, 1;

-105*t^4, -75*t^3, -30*t^2, -10*t, -5, 1;

Let Pc(n,t) = LPT(Pb(.,t)).

Then [TB(t)]^(-1) = TC(t) = [ binomial(n,k) * Pc(n-k,t) ] = LPT(TB),

whose first column is

Pc(0,t) = 1

Pc(1,t) = 1

Pc(2,t) = 2 + t

Pc(3,t) = 6 + 6*t + 3*t^2

Pc(4,t) = 24 + 36*t + 30*t^2 + 15*t^3

Pc(5,t) = 120 + 240*t + 270*t^2 + 210*t^3 + 105*t^4.

The coefficients of these polynomials are given by the reverse of A102625 with the highest order coefficients given by A001147 with an additional leading 1.

Note this is not the complete matrix TC. The complete matrix is formed by multiplying along the diagonal of the lower triangular Pascal matrix by these polynomials, embedding trees of coefficients in the matrix.

exp[Pb(.,t)*x] = 1 + [(1-2t*x)^(1/2) - 1] / (t-0) = [1 + a finite diff. of [(1-2t*x)^(1/2)] with step t] = e.g.f. of the first column of TB.

exp[Pc(.,t)*x] = 1 / { 1 + [(1-2t*x)^(1/2) - 1] / t } = 1 / exp[Pb(.,t)*x) = e.g.f. of the first column of TC.

TB(t) and TC(t), being inverse to each other, are the generators of an Abelian group.

TB(0) and TC(0) are generators for a subgroup representing the iterated Laguerre operator described in A132013 and A132014.

Let sb(t,m) and sc(t,m) be the associated sequences under the LPT to TB(t)^m = B(t,m) and TC(t)^m = C(t,m).

Let Esb(t,m) and Esc(t,m) be e.g.f.'s for sb(t,m) and sc(t,m), rB(t,m) and rC(t,m) be the row sums of B(t,m) and C(t,m) and aB(t,m) and aC(t,m) be the alternating row sums.

Then B(t,m) is the inverse of C(t,m), Esb(t,m) is the reciprocal of Esc(t,m) and sb(t,m) and sc(t,m) form a reciprocal pair under the LPT. Similar relations hold among the row sums and the alternating sign row sums and associated quantities.

All the group members have the form B(t,m) * C(u,p) = TB(t)^m * TC(u)^p = [ binomial(n,k) * s(n-k) ]

with associated e.g.f. Es(x) = exp[m * Pb(.,t) * x] * exp[p * Pc(.,u) * x] for the first column of the matrix, with terms s(n), so group multiplication is isomorphic to matrix multiplication and to multiplication of the e.g.f.'s for the associated sequences (see examples).

These results can be extended to other groups of integer-valued arrays by replacing the 2 by any natural number in the expression for exp[Pb(.,t)*x].

More generally,

[ G.f. for M = Product_{i=0..j} B[s(i),m(i)] * C[t(i),n(i)] ]

= exp(u*x) * Product_{i=0..j} { exp[m(i) * Pb(.,s(i)) * x] * exp[n(i) * Pc(.,t(i)) * x] }

= exp(u*x) * Product_{i=0..j} { 1 + [ (1 - 2*s(i)*x)^(1/2) - 1 ] / s(i) }^m(i) / { 1 + [ (1 - 2*t(i)*x)^(1/2) - 1 ] / t(i) }^n(i)

= exp(u*x) * H(x)

[ E.g.f. for M ] = I_o[2*(u*x)^(1/2)] * H(x).

M is an integer-valued matrix for m(i) and n(i) positive integers and s(i) and t(i) integers. To invert M, change B to C in Product for M.

H(x) is the e.g.f. for the first column of M and diagonally multiplying the Pascal matrix by the terms of this column generates M. See examples.

The G.f. for M, i.e., the e.g.f. for the row polynomials of M, implies that the row polynomials form an Appell sequence (see Wikipedia and Mathworld). - Tom Copeland, Dec 03 2013

Here two n-gons are said to be equivalent if they differ in starting point, orientation, or by a rotation (but not by a reflection - for that see A000940).

Number of cycle necklaces on n vertices, defined as equivalence classes of (labeled, undirected) Hamiltonian cycles under rotation of the vertices. The path version is A275527. - Gus Wiseman, Mar 02 2019

From Sanjar Abrarov, Mar 30 2019: (Start)

There is a formula for numerical integration (see MATLAB Central file ID# 71037):

Integral_{x=0..1} f(x) dx = 2*Sum_{m=1..M} Sum_{n>=0} 1/((2*M)^(2*n + 1)*(2*n + 1)!)*f^(2*n)(x)|_x = (m - 1/2)/M, where the notation f^(2*n)(x)|_x = (m - 1/2)/M is the (2*n)-th derivative of the function f(x) at the points x = (m - 1/2)/M.

When we choose M = 1, then the corresponding coefficients are generated as 2*1/(2^(2*n + 1)*(2*n + 1)!) = 1/(4^n*(2*n + 1)!).

Therefore, this sequence also occurs in the denominator of the numerical integration formula at M = 1. (End)

From Peter Bala, Oct 03 2019: (Start)

Denominators in the expansion of 2*sinh(x/2) = x + x^3/24 + x^5/1920 + x^7/322560 + ....

If f(x) is a polynomial in x then the central difference f(x+1/2) - f(x-1/2) = 2*sinh(D/2)(f(x)) = D(f(x)) + (1/24)*D^3(f(x)) + (1/1920)*D^5(f(x)) + ..., where D denotes the differential operator d/dx. Formulas for higher central differences in terms of powers of the operator D can be obtained from the expansion of powers of the function 2*sinh(x/2). For example, the expansion (2*sinh(x/2))^2 = x^2 + (1/12)*x^4 + (1/360)*x^6 + .. leads to the second central difference formula f(x+1) - 2*f(x) + f(x-1) = D^2(f(x)) + (1/12)*D^4(f(x)) + (1/360)* D^6(f(x)) + .... See A002674. (End)