cp's OEIS Frontend

This version indexes the Eulerian numbers in the same way as Graham et al.'s Concrete Mathematics (see references section). The traditional indexing, used by Riordan, Comtet and others, is given in A008292, which is the main entry for the Eulerian numbers.

Each row of A123125 is the reverse of the corresponding row in A173018. - Michael Somos Mar 17 2011

Triangle T(n,k), read by rows, given by [1,0,2,0,3,0,4,0,5,0,6,0,7,0,8,0,9,...] DELTA [0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,8,0,9,...] where DELTA is the operator defined in A084938. - Philippe Deléham Sep 30 2011

[ E(.,t)/(1-t)]^n = n!*Lag[n,-P(.,t)/(1-t)] and [ -P(.,t)/(1-t)]^n = n!*Lag[n, E(.,t)/(1-t)] umbrally comprise a combinatorial Laguerre transform pair, where E(n,t) are the Eulerian polynomials (e.g., E(2,t)= 1+t) and P(n,t) are the polynomials related to polylogarithms in A131758. - Tom Copeland, Oct 03 2014

See A131758 for connections of the evaluation of these polynomials at -1 (alternating row sum) to the Euler, Genocchi, Bernoulli, and zag/tangent numbers and values of the Riemann zeta function and polylogarithms. See also A119879 for the Swiss-knife polynomials. - Tom Copeland, Oct 20 2015

The beginning of this sequence does not quite agree with the usual version, which is A173018. - N. J. A. Sloane, Nov 21 2010

Each row of A123125 is the reverse of the corresponding row in A173018. - Michael Somos, Mar 17 2011

A008292 (subtriangle for k>=1 and n>=1) is the main entry for these numbers.

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,...] DELTA [1,0,2,0,3,0,4,0,5,0,6,...] where DELTA is the operator defined in A084938.

Row sums are the factorials. - Roger L. Bagula and Gary W. Adamson, Aug 14 2008

If the initial zero column is deleted, the result is A008292. - Roger L. Bagula and Gary W. Adamson, Aug 14 2008

This result gives an alternative method of calculating the Eulerian numbers by an Umbral Calculus expansion from Comtet. - Roger L. Bagula, Nov 21 2009

This function seems to be equivalent to the PolyLog expansion. - Roger L. Bagula, Nov 21 2009

A raising operator formed from the e.g.f. of this entry is the generator of a sequence of polynomials p(n,x;t) defined in A046802 that specialize to those for A119879 as p(n,x;-1), A007318 as p(n,x;0), A073107 as p(n,x;1), and A046802 as p(n,0;t). See Copeland link for more associations. - Tom Copeland, Oct 20 2015

The Eulerian numbers in this setup count the permutation trees of power n and width k (see the Luschny link). For the associated combinatorial statistic over permutations see the Sage program below and the example section. - Peter Luschny, Dec 09 2015 [See Elder et al. link. Peter Luschny, Jul 13 2022]

From Wolfdieter Lang, Apr 03 2017: (Start)

The row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k are the numerator polynomials of the o.g.f. G(n, x) of n-powers {m^n}_{m>=0} (with 0^0 = 1): G(n, x) = R(n, x)/(1-x)^(n+1). See the Aug 14 2008 formula, where f(x,n) = R(n, x). The e.g.f. of R(n, t) is given in Copeland's Oct 14 2015 formula below.

The first nine column sequences are A000007, A000012, A000295, A000460, A000498, A000505, A000514, A001243, A001244. (End)

With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of this entry, A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of A248727 (the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - Tom Copeland, Jan 24 2020

Let b(n) = (1/(n+1))*Sum_{k=0..n-1} (-1)^(n-k+1)*T(n, k+1) / binomial(n, k+1). Then b(n) = Bernoulli(n, 1) = -n*Zeta(1 - n) = Integral_{x=0..1} F_n(x) for n >= 1. Here F_n(x) are the signed Fubini polynomials (A278075). (See also Rzadkowski and Urlinska, example 1.) - Peter Luschny, Feb 15 2021

Patrick J. Burchell (see link) describes the following method: To get the k-th row of the triangle write the nonnegative integers with a fixed exponent k as a sequence, 0^k, 1^k, 2^k, ..., and then apply the first differences to them k + 1 times. - Peter Luschny, Apr 02 2023

Let M = n X n matrix with (i,j)-th entry a(n+1-j, n+1-i), e.g., if n = 3, M = [1 1 1; 3 1 0; 2 0 0]. Given a sequence s = [s(0)..s(n-1)], let b = [b(0)..b(n-1)] be its inverse binomial transform and let c = [c(0)..c(n-1)] = M^(-1)*transpose(b). Then s(k) = Sum_{i=0..n-1} b(i)*binomial(k,i) = Sum_{i=0..n-1} c(i)*k^i, k=0..n-1. - Gary W. Adamson, Nov 11 2001

From Gary W. Adamson, Aug 09 2008: (Start)

Julius Worpitzky's 1883 algorithm generates Bernoulli numbers.

By way of example [Wikipedia]:

B0 = 1;

B1 = 1/1 - 1/2;

B2 = 1/1 - 3/2 + 2/3;

B3 = 1/1 - 7/2 + 12/3 - 6/4;

B4 = 1/1 - 15/2 + 50/3 - 60/4 + 24/5;

B5 = 1/1 - 31/2 + 180/3 - 390/4 + 360/5 - 120/6;

B6 = 1/1 - 63/2 + 602/3 - 2100/4 + 3360/5 - 2520/6 + 720/7;

...

Note that in this algorithm, odd n's for the Bernoulli numbers sum to 0, not 1, and the sum for B1 = 1/2 = (1/1 - 1/2). B3 = 0 = (1 - 7/2 + 13/3 - 6/4) = 0. The summation for B4 = -1/30. (End)

Pursuant to Worpitzky's algorithm and given M = A028246 as an infinite lower triangular matrix, M * [1/1, -1/2, 1/3, ...] (i.e., the Harmonic series with alternate signs) = the Bernoulli numbers starting [1/1, 1/2, 1/6, ...]. - Gary W. Adamson, Mar 22 2012

From Tom Copeland, Oct 23 2008: (Start)

G(x,t) = 1/(1 + (1-exp(x*t))/t) = 1 + 1 x + (2 + t)*x^2/2! + (6 + 6t + t^2)*x^3/3! + ... gives row polynomials for A090582, the f-polynomials for the permutohedra (see A019538).

G(x,t-1) = 1 + 1*x + (1 + t)*x^2 / 2! + (1 + 4t + t^2)*x^3 / 3! + ... gives row polynomials for A008292, the h-polynomials for permutohedra.

G[(t+1)x,-1/(t+1)] = 1 + (1+ t) x + (1 + 3t + 2 t^2) x^2 / 2! + ... gives row polynomials for the present triangle. (End)

The Worpitzky triangle seems to be an apt name for this triangle. - Johannes W. Meijer, Jun 18 2009

If Pascal's triangle is written as a lower triangular matrix and multiplied by A028246 written as an upper triangular matrix, the product is a matrix where the (i,j)-th term is (i+1)^j. For example,

1,0,0,0 1,1,1, 1 1,1, 1, 1

1,1,0,0 * 0,1,3, 7 = 1,2, 4, 8

1,2,1,0 0,0,2,12 1,3, 9,27

1,3,3,1 0,0,0, 6 1,4,16,64

So, numbering all three matrices' rows and columns starting at 0, the (i,j) term of the product is (i+1)^j. - Jack A. Cohen (ProfCohen(AT)comcast.net), Aug 03 2010

The Fi1 and Fi2 triangle sums are both given by sequence A000670. For the definition of these triangle sums see A180662. The mirror image of the Worpitzky triangle is A130850. - Johannes W. Meijer, Apr 20 2011

Let S_n(m) = 1^m + 2^m + ... + n^m. Then, for n >= 0, we have the following representation of S_n(m) as a linear combination of the binomial coefficients:

S_n(m) = Sum_{i=1..n+1} a(i+n*(n+1)/2)*C(m,i). E.g., S_2(m) = a(4)*C(m,1) + a(5)*C(m,2) + a(6)*C(m,3) = C(m,1) + 3*C(m,2) + 2*C(m,3). - Vladimir Shevelev, Dec 21 2011

Given the set X = [1..n] and 1 <= k <= n, then a(n,k) is the number of sets T of size k of subset S of X such that S is either empty or else contains 1 and another element of X and such that any two elemements of T are either comparable or disjoint. - Michael Somos, Apr 20 2013

Working with the row and column indexing starting at -1, a(n,k) gives the number of k-dimensional faces in the first barycentric subdivision of the standard n-dimensional simplex (apply Brenti and Welker, Lemma 2.1). For example, the barycentric subdivision of the 2-simplex (a triangle) has 1 empty face, 7 vertices, 12 edges and 6 triangular faces giving row 4 of this triangle as (1,7,12,6). Cf. A053440. - Peter Bala, Jul 14 2014

See A074909 and above g.f.s for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - Tom Copeland, Nov 14 2014

An e.g.f. G(x,t) = exp[P(.,t)x] = 1/t - 1/[t+(1-t)(1-e^(-xt^2))] = (1-t) * x + (-2t + 3t^2 - t^3) * x^2/2! + (6t^2 - 12t^3 + 7t^4 - t^5) * x^3/3! + ... for the shifted, reverse, signed polynomials with the first element nulled, is generated by the infinitesimal generator g(u,t)d/du = [(1-u*t)(1-(1+u)t)]d/du, i.e., exp[x * g(u,t)d/du] u eval. at u=0 generates the polynomials. See A019538 and the G. Rzadkowski link below for connections to the Bernoulli and Eulerian numbers, a Ricatti differential equation, and a soliton solution to the KdV equation. The inverse in x of this e.g.f. is Ginv(x,t) = (-1/t^2)*log{[1-t(1+x)]/[(1-t)(1-tx)]} = [1/(1-t)]x + [(2t-t^2)/(1-t)^2]x^2/2 + [(3t^2-3t^3+t^4)/(1-t)^3]x^3/3 + [(4t^3-6t^4+4t^5-t^6)/(1-t)^4]x^4/4 + ... . The numerators are signed, shifted A135278 (reversed A074909), and the rational functions are the columns of A074909. Also, dG(x,t)/dx = g(G(x,t),t) (cf. A145271). (Analytic G(x,t) added, and Ginv corrected and expanded on Dec 28 2015.) - Tom Copeland, Nov 21 2014

The operator R = x + (1 + t) + t e^{-D} / [1 + t(1-e^(-D))] = x + (1+t) + t - (t+t^2) D + (t+3t^2+2t^3) D^2/2! - ... contains an e.g.f. of the reverse row polynomials of the present triangle, i.e., A123125 * A007318 (with row and column offset 1 and 1). Umbrally, R^n 1 = q_n(x;t) = (q.(0;t)+x)^n, with q_m(0;t) = (t+1)^(m+1) - t^(m+1), the row polynomials of A074909, and D = d/dx. In other words, R generates the Appell polynomials associated with the base sequence A074909. For example, R 1 = q_1(x;t) = (q.(0;t)+x) = q_1(0;t) + q__0(0;t)x = (1+2t) + x, and R^2 1 = q_2(x;t) = (q.(0;t)+x)^2 = q_2(0:t) + 2q_1(0;t)x + q_0(0;t)x^2 = 1+3t+3t^2 + 2(1+2t)x + x^2. Evaluating the polynomials at x=0 regenerates the base sequence. With a simple sign change in R, R generates the Appell polynomials associated with A248727. - Tom Copeland, Jan 23 2015

For a natural refinement of this array, see A263634. - Tom Copeland, Nov 06 2015

From Wolfdieter Lang, Mar 13 2017: (Start)

The e.g.f. E(n, x) for {S(n, m)}{m>=0} with S(n, m) = Sum{k=1..m} k^n, n >= 0, (with undefined sum put to 0) is exp(x)*R(n+1, x) with the exponential row polynomials R(n, x) = Sum_{k=1..n} a(n, k)*x^k/k!. E.g., e.g.f. for n = 2, A000330: exp(x)*(1*x/1!+3*x^2/2!+2*x^3/3!).

The o.g.f. G(n, x) for {S(n, m)}{m >=0} is then found by Laplace transform to be G(n, 1/p) = p*Sum{k=1..n} a(n+1, k)/(p-1)^(2+k).

Hence G(n, x) = x/(1 - x)^(n+2)*Sum_{k=1..n} A008292(n,k)*x^(k-1).

E.g., n=2: G(2, 1/p) = p*(1/(p-1)^2 + 3/(p-1)^3 + 2/(p-1)^4) = p^2*(1+p)/(p-1)^4; hence G(2, x) = x*(1+x)/(1-x)^4.

This works also backwards: from the o.g.f. to the e.g.f. of {S(n, m)}_{m>=0}. (End)

a(n,k) is the number of k-tuples of pairwise disjoint and nonempty subsets of a set of size n. - Dorian Guyot, May 21 2019

From Rajesh Kumar Mohapatra, Mar 16 2020: (Start)

a(n-1,k) is the number of chains of length k in a partially ordered set formed from subsets of an n-element set ordered by inclusion such that the first term of the chains is either the empty set or an n-element set.

Also, a(n-1,k) is the number of distinct k-level rooted fuzzy subsets of an n-set ordered by set inclusion. (End)

The relations on p. 34 of Hasan (also p. 17 of Franco and Hasan) agree with the relation between A019538 and this entry given in the formula section. - Tom Copeland, May 14 2020

T(n,k) is the size of the Green's L-classes in the D-classes of rank (k-1) in the semigroup of partial transformations on an (n-1)-set. - Geoffrey Critzer, Jan 09 2023

T(n,k) is the number of strongly connected binary relations on [n] that have period k (A367948) and index 1. See Theorem 5.4.25(6) in Ki Hang Kim reference. - Geoffrey Critzer, Dec 07 2023

Previous name was: a(n) = Sum_{k=0..n-1} (-1)^(k)*C(n-1,k)*a(n-1-k)*a(k) for n>0 with a(0)=1.

Factorials have a similar recurrence: f(n) = Sum_{k=0..n-1} C(n-1,k)*f(n-1-k)*f(k), n > 0.

Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times Sum_{k=0..n} binomial(n,k)*k^q, then A155585(x) = Sum_{k=0..x-1} T(x,k)*(-1)^k. See Mathematica code below. - John M. Campbell, Nov 16 2011

For the difference table and the relation to the Seidel triangle see A239005. - Paul Curtz, Mar 06 2014

From Tom Copeland, Sep 29 2015: (Start)

Let z(t) = 2/(e^(2t)+1) = 1 + tanh(-t) = e.g.f.(-t) for this sequence = 1 - t + 2*t^3/3! - 16*t^5/5! + ... .

dlog(z(t))/dt = -z(-t), so the raising operators that generate Appell polynomials associated with this sequence, A081733, and its reciprocal, A119468, contain z(-d/dx) = e.g.f.(d/dx) as the differential operator component.

dz(t)/dt = z*(z-2), so the assorted relations to a Ricatti equation, the Eulerian numbers A008292, and the Bernoulli numbers in the Rzadkowski link hold.

From Michael Somos's formula below (drawing on the Edwards link), y(t,1)=1 and x(t,1) = (1-e^(2t))/(1+e^(2t)), giving z(t) = 1 + x(t,1). Compare this to the formulas in my list in A008292 (Sep 14 2014) with a=1 and b=-1,

A) A(t,1,-1) = A(t) = -x(t,1) = (e^(2t)-1)/(1+e^(2t)) = tanh(t) = t + -2*t^3/3! + 16*t^5/5! + -272*t^7/7! + ... = e.g.f.(t) - 1 (see A000182 and A000111)

B) Ainv(t) = log((1+t)/(1-t))/2 = tanh^(-1)(t) = t + t^3/3 + t^5/5 + ..., the compositional inverse of A(t)

C) dA/dt = (1-A^2), relating A(t) to a Weierstrass elliptic function

D) ((1-t^2)d/dt)^n t evaluated at t=0, a generator for the sequence A(t)

F) FGL(x,y)= (x+y)/(1+xy) = A(Ainv(x) + Ainv(y)), a related formal group law corresponding to the Lorentz FGL (Lorentz transformation--addition of parallel velocities in special relativity) and the Atiyah-Singer signature and the elliptic curve (1-t^2)*s = t^3 in Tate coordinates according to the Lenart and Zainoulline link and the Buchstaber and Bunkova link (pp. 35-37) in A008292.

A133437 maps the reciprocal odd natural numbers through the refined faces of associahedra to a(n).

A145271 links the differential relations to the geometry of flow maps, vector fields, and thereby formal group laws. See Mathworld for links of tanh to other geometries and statistics.

Since the a(n) are related to normalized values of the Bernoulli numbers and the Riemann zeta and Dirichlet eta functions, there are links to Witten's work on volumes of manifolds in two-dimensional quantum gauge theories and the Kervaire-Milnor formula for homotopy groups of hyperspheres (see my link below).

See A101343, A111593 and A059419 for this and the related generator (1 + t^2) d/dt and associated polynomials. (End)

With the exception of the first term (1), entries are the alternating sums of the rows of the Eulerian triangle, A008292. - Gregory Gerard Wojnar, Sep 29 2018

In the following the expression [n odd] is 1 if n is odd, 0 otherwise.

(+) W_n(0) = E_n are the Euler (or secant) numbers A122045.

(+) W_n(1) = T_n are the signed tangent numbers, see A009006.

(+) W_{n-1}(1) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli number A027641/A027642.

(+) W_n(-1) 2^{-n}(n+1) = G_n the Genocchi number A036968.

(+) W_n(1/2) 2^{n} are the signed generalized Euler (Springer) number, see A001586.

(+) | W_n([n odd]) | the number of alternating permutations A000111.

(+) | W_n([n odd]) / n! | for 0<=n the Euler zeta number A099612/A099617 (see Wikipedia on Bernoulli number). - Peter Luschny, Dec 29 2008

The diagonals in the full triangle (with zero coefficients) of the polynomials have the general form E(k)*binomial(n+k,k) (k>=0 fixed, n=0,1,...) where E(n) are the Euler numbers in the enumeration A122045. For k=2 we find the triangular numbers A000217 and for k=4 A154286. - Peter Luschny, Jan 06 2009

From Peter Bala, Jun 10 2009: (Start)

The Swiss-Knife polynomials W_n(x) may be expressed in terms of the Bernoulli polynomials B(n,x) as

... W_n(x) = 4^(n+1)/(2*n+2)*[B(n+1,(x+3)/4) - B(n+1,(x+1)/4)].

The Swiss-Knife polynomials are, apart from a multiplying factor, examples of generalized Bernoulli polynomials.

Let X be the Dirichlet character modulus 4 defined by X(4*n+1) = 1, X(4*n+3) = -1 and X(2*n) = 0. The generalized Bernoulli polynomials B(X;n,x), n = 1,2,..., associated with the character X are defined by means of the generating function

... t*exp(x*t)*(exp(t)-exp(3*t))/(exp(4*t)-1) = sum {n = 1..inf} B(X;n,x)*t^n/n!.

The first few values are B(X;1,x) = -1/2, B(X;2,x) = -x, B(X,3,x) = -3/2*(x^2-1) and B(X;4,x) = -2*(x^3-3*x).

In general, W_n(x) = -2/(n+1)*B(X;n+1,x).

For the theory of generalized Bernoulli polynomials associated to a periodic arithmetical function see [Cohen, Section 9.4].

The generalized Bernoulli polynomials may be used to evaluate twisted sums of k-th powers. For the present case the result is

sum{n = 0..4*N-1} X(n)*n^k = 1^k - 3^k + 5^k - 7^k + ... - (4*N-1)^k

= [B(X;k+1,4*N) - B(X;k+1,0)]/(k+1) = [W_k(0) - W_k(4*N)]/2.

For the proof apply [Cohen, Corollary 9.4.17 with m = 4 and x = 0].

The generalized Bernoulli polynomials and the Swiss-Knife polynomials are also related to infinite sums of powers through their Fourier series - see the formula section below. For a table of the coefficients of generalized Bernoulli polynomials attached to a Dirichlet character modulus 8 see A151751.

(End)

The Swiss-Knife polynomials provide a general formula for alternating sums of powers similar to the formula which are provided by the Bernoulli polynomials for non-alternating sums of powers (see the Luschny link). Sequences covered by this formula include A001057, A062393, A062392, A011934, A144129, A077221, A137501, A046092. - Peter Luschny, Jul 12 2009

The greatest common divisor of the nonzero coefficients of the decapitated Swiss-Knife polynomials is exp(Lambda(n)), where Lambda(n) is the von Mangoldt function for odd primes, symbolically:

gcd(coeffs(SKP_{n}(x) - x^n)) = A155457(n) (n>1). - Peter Luschny, Dec 16 2009

Another version is at A119879. - Philippe Deléham, Oct 26 2013

T(n,k) is the number of positroid cells of the totally nonnegative Grassmannian G+(k,n) (cf. Postnikov/Williams). It is the triangle of the h-vectors of the stellahedra. - Tom Copeland, Oct 10 2014

See A248727 for a simple transformation of the row polynomials of this entry that produces the umbral compositional inverses of the polynomials of A074909, related to the face polynomials of the simplices. - Tom Copeland, Jan 21 2015

From Tom Copeland, Jan 24 2015: (Start)

The reciprocal of this entry's e.g.f. is [t e^(-xt) - e^(-x)] / (t-1) = 1 - (1+t) x + (1+t+t^2) x^2/2! - (1+t+t^2+t^3) x^3/3! + ... = e^(q.(0;t)x), giving the base sequence (q.(0;t))^n = q_n(0;t) = (-1)^n [1-t^(n+1)] / (1-t) for the umbral compositional inverses (q.(0;t)+z)^n = q_n(z;t) of the Appell polynomials associated with this entry, p_n(z;t) below, i.e., q_n(p.(z;t)) = z^n = p_n(q.(z;t)), in umbral notation. The relations in A133314 then apply between the two sets of base polynomials. (Inserted missing index in a formula - Mar 03 2016.)

The associated o.g.f. for the umbral inverses is Og(x) = x / (1-x q.(0:t)) = x / [(1+x)(1+tx)] = x / [1+(1+t)x+tx^2]. Applying A134264 to h(x) = x / Og(x) = 1 + (1+t) x + t x^2 leads to an o.g.f. for the Narayana polynomials A001263 as the comp. inverse Oginv(x) = [1-(1+t)x-sqrt[1-2(1+t)x+((t-1)x)^2]] / (2xt). Note that Og(x) gives the signed h-polynomials of the simplices and that Oginv(x) gives the h-polynomials of the simplicial duals of the Stasheff polynomials, or type A associahedra. Contrast this with A248727 = A046802 * A007318, which has o.g.f.s related to the corresponding f-polynomials. (End)

The Appell polynomials p_n(x;t) in the formulas below specialize to the Swiss-knife polynomials of A119879 for t = -1, so the Springer numbers A001586 are given by 2^n p_n(1/2;-1). - Tom Copeland, Oct 14 2015

The row polynomials are the h-polynomials associated to the stellahedra, whose f-polynomials are the row polynomials of A248727. Cf. page 60 of the Buchstaber and Panov link. - Tom Copeland, Nov 08 2016

The row polynomials are the h-polynomials of the stellohedra, which enumerate partial permutations according to descents. Cf. Section 10.4 of the Postnikov-Reiner-Williams reference. - Lauren Williams, Jul 05 2022

From p. 60 of the Buchstaber and Panov link, S = P * C / T where S, P, C, and T are the bivariate e.g.f.s of the h vectors of the stellahedra, permutahedra, hypercubes, and (n-1)-simplices, respectively. - Tom Copeland, Jan 09 2017

The number of Le-diagrams of type (k, n) this means the diagram uses the bounding box size k x (n-k), equivalently the number of Grassmann necklaces of type (k, n) and also the number of decorated permutations with k anti-exceedances. - Thomas Scheuerle, Dec 29 2024

Let Q(m, n) = Sum_(k=0..n-1) (-1)^k * binomial(n, k) * (n-k)^m. Then Q(m,n) is the numerator of the probability P(m,n) = Q(m,n)/n^m of seeing each card at least once if m >= n cards are drawn with replacement from a deck of n cards, written in a two-dimensional array read by antidiagonals with Q(m,m) as first row and Q(m,1) as first column.

The sequence is given as a matrix with the first row containing the cases #draws = size_of_deck. The second row contains #draws = 1 + size_of_deck. If "mn" indicates m cards drawn from a deck with n cards then the locations in the matrix are:

11 22 33 44 55 66 77 ...

21 32 43 54 65 76 87 ...

31 42 53 64 75 86 97 ...

41 52 63 74 85 .. .. ...

read by antidiagonals ->:

11, 22, 21, 33, 32, 31, 44, 43, 42, 41, 55, 54, 53, 52, ....

The probabilities are given by Q(m,n)/n^m:

.(m,n):.....11..22..21..33..32..31..44..43..42..41...55...54..53..52..51

.....Q:......1...2...1...6...6...1..24..36..14...1..120..240.150..30...1

...n^m:......1...4...1..27...8...1.256..81..16...1.3125.1024.243..32...1

%.Success:.100..50.100..22..75.100...9..44..88.100....4...23..62..94.100

P(n,n) = n!/n^n which can be approximated by sqrt(Pi*(2n+1/3))/e^n (Gosper's approximation to n!).

Let P[n] be the set of all n-permutations. Build a superset Q[n] of P[n] composed of n-permutations in which some (possibly all or none) ascents have been designated. An ascent in a permutation s[1]s[2]...s[n] is a pair of consecutive elements s[i],s[i+1] such that s[i] < s[i+1]. As a triangular array read by rows T(n,k) is the number of elements in Q[n] that have exactly k distinguished ascents, n >= 1, 0 <= k <= n-1. Row sums are A000670. E.g.f. is y/(1+y-exp(y*x)). For example, T(3,1)=6 because there are four 3-permutations with one ascent, with these we would also count 1->2 3, and 1 2->3 where exactly one ascent is designated by "->". (After Flajolet and Sedgewick.) - Geoffrey Critzer, Nov 13 2012

Sum_(k=1..n) Q(n, k)*binomial(x, k) = x^n such that Sum_{k=1..n} Q(n, i)*binomial(x+1,i+1) = Sum_{k=1..x} k^n. - David A. Corneth, Feb 17 2014

A141618(n,n-k+1) = a(n,k) * C(n,k-1) / k. - Tom Copeland, Oct 25 2014

See A074909 and above g.f.s below for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - Tom Copeland, Nov 14 2014

For connections to toric varieties and Eulerian polynomials (in addition to those noted below), see the Dolgachev and Lunts and the Stembridge links in A019538. - Tom Copeland, Dec 31 2015

See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra (this entry) and stellahedra. - Tom Copeland, Nov 14 2016

From the Hasan and Franco and Hasan papers: The m-permutohedra for m=1,2,3,4 are the line segment, hexagon, truncated octahedron and omnitruncated 5-cell. The first three are well-known from the study of elliptic models, brane tilings and brane brick models. The m+1 torus can be tiled by a single (m+2)-permutohedron. Relations to toric Calabi-Yau Kahler manifolds are also discussed. - Tom Copeland, May 14 2020

Row sums are A011782. Diagonal sums are F(n+1)*(1+(-1)^n)/2 (aerated version of A001519). Product by Pascal's triangle A007318 is A119468. Schur product of (1/(1-x),x/(1-x)) and (1/(1-x^2),x).

Exponential Riordan array (cosh(x),x). Inverse is (sech(x),x) or A119879. - Paul Barry, May 26 2006

Rows give coefficients of polynomials p_n(x) = Sum_{k=0..n} (k+1 mod 2)*binomial(n,k)*x^(n-k) having e.g.f. exp(x*t)*cosh(t)= 1*(t^0/0!) + x*(t^1/1!) + (1+x^2)*(t^2/2!) + ... - Peter Luschny, Jul 14 2009

Inverse of the coefficient matrix of the Swiss-Knife polynomials in ascending order of x^i (reversed and aerated rows of A153641). - Peter Luschny, Jul 16 2012

Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0 M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*... is equal to A136630 but with the first row and column omitted. - Peter Bala, Jul 28 2014

The row polynomials SKv(n,x) = [(x+1)^n + (x-1)^n]/2 , with e.g.f. cosh(t)*exp(xt), are the umbral compositional inverses of the row polynomials of A119879 (basically the Swiss Knife polynomials SK(n,x) of A153641); i.e., umbrally SKv(n,SK(.,x)) = x^n = SK(n,SKv(.,x)). Therefore, this entry's matrix and A119879 are an inverse pair. Both sequences of polynomials are Appell sequences, i.e., d/dx P(n,x) = n * P(n-1,x) and (P(.,x)+y)^n = P(n,x+y). In particular, (SKv(.,0)+x)^n = SKv(n,x), reflecting that the first column has the e.g.f. cosh(t). The raising operator is R = x + tanh(d/dx); i.e., R SKv(n,x) = SKv(n+1,x). The coefficients of this operator are basically the signed and aerated zag numbers A000182, which can be expressed as normalized Bernoulli numbers. The triangle is formed by multiplying the n-th diagonal of the lower triangular Pascal matrix by the Taylor series coefficient a(n) of cosh(x). More relations for this type of triangle and its inverse are given by the formalism of A133314. - Tom Copeland, Sep 05 2015

The signed version of this matrix has the e.g.f. cos(t) e^{xt}, generating Appell polynomials that have only real, simple zeros and whose extrema are maxima above the x-axis and minima below and situated above and below the zeros of the next lower degree polynomial. The bivariate versions appear on p. 27 of Dimitrov and Rusev in conditions for entire functions that are cosine transforms of a class of functions to have only real zeros. - Tom Copeland, May 21 2020

The n-th row of the triangle is obtained by multiplying by 2^(n-1) the elements of the first row of the limit as k approaches infinity of the stochastic matrix P^(2k-1) where P is the stochastic matrix associated with the Ehrenfest model with n balls. The elements of a stochastic matrix P give the probabilities of arriving in a state j given the previous state i. In particular the sum of every row of the matrix must be 1, and so the sum of the terms of the n-th row of this triangle is 2^(n-1). Furthermore, by the properties of Markov chains, we can interpret P^(2k-1) as the (2k-1)-step transition matrix of the Ehrenfest model and its limit exists and it is again a stochastic matrix. The rows of the triangle divided by 2^(n-1) are the even rows (second, fourth, ...) and the odd rows (first, third, ...) of the limit matrix P^(2k-1). - Luca Onnis, Oct 29 2023

This is a transform of A046802 treating it as an array of h-vectors, so y is replaced by (y+1) in the e.g.f. for A046802.

An e.g.f. for the reversed row polynomials with signs is given by exp(a.(0;t)x) = [e^{(1+t)x} [1+t(1-e^(-x))]]^(-1) = 1 - (1+2t)x + (1+5t+5t^2)x^2/2! + ... . The reciprocal is an e.g.f. for the reversed face polynomials of the simplices A074909, i.e., exp(b.(0;t)x) = e^{(1+t)x} [1+t(1-e^(-x))] = 1 + (1+2t)x +(1+3t+3t^2) x^2/2! + ... , so the relations of A133314 apply between the two sets of polynomials. In particular, umbrally [a.(0;t)+b.(0;t)]^n vanishes except for n=0 for which it's unity, implying the two sets of Appell polynomials formed from the two bases, a_n(z;t) = (a.(0;t)+z)^n and b_n(z;t) = (b.(0;t) + z)^n, are an umbral compositional inverse pair, i.e., b_n(a.(x;t);t)= x^n = a_n(b.(x;t);t). Raising operators for these Appell polynomials are related to the polynomials of A028246, whose reverse polynomials are given by A123125 * A007318. Compare: A248727 = A007318 * A123125 * A007318 and A046802 = A007318 * A123125. See A074909 for definitions and related links. - Tom Copeland, Jan 21 2015

The o.g.f. for the umbral inverses is Og(x) = x / (1 - x b.(0;t)) = x / [(1-tx)(1-(1+t)x)] = x + (1+2t) x^2 + (1+3t+3t^2) x^3 + ... . Its compositional inverse is an o.g.f for signed A033282, the reverse f-polynomials for the simplicial duals of the Stasheff polytopes, or associahedra of type A, Oginv(x) =[1+(1+2t)x-sqrt[1+2(1+2t)x+x^2]] / (2t(1+t)x) = x - (1+2t) x^2 + (1+5t+5t^2) x^3 + ... . Contrast this with the o.g.f.s related to the corresponding h-polynomials in A046802. - Tom Copeland, Jan 24 2015

Face vectors, or coefficients of the face polynomials, of the stellahedra, or stellohedra. See p. 59 of Buchstaber and Panov. - Tom Copeland, Nov 08 2016

See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra and stellahedra. - Tom Copeland, Nov 14 2016

Transform of 2^n under the matrix A119879.

Also the Swiss-Knife polynomials A153641 evaluated at x=2. - Peter Luschny, Nov 23 2012