cp's OEIS Frontend

These polynomials arise in the following setup. Suppose G and H are power series satisfying G + H = G*H = 1/x. Then G^n + H^n = (1/x^n)*L_n(-x).

Apart from signs, triangle of coefficients when 2*cos(nt) is expanded in terms of x = 2*cos(t). For example, 2*cos(2t) = x^2 - 2, 2*cos(3t) = x^3 - 3x and 2*cos(4t) = x^4 - 4x^2 + 2. - Anthony C Robin, Jun 02 2004

Triangle of coefficients of expansion of Z_{nk} in terms of Z_k.

Row n has 1 + floor(n/2) terms. - Emeric Deutsch, Dec 25 2004

T(n,k) = number of k-matchings of the cycle C_n (n > 1). Example: T(6,2)=9 because the 2-matchings of the hexagon with edges a, b, c, d, e, f are ac, ad, ae, bd, be, bf, ce, cf and df. - Emeric Deutsch, Dec 25 2004

An example for the first comment: G=c(x), H=1/(x*c(x)) with c(x) the o.g.f. Catalan numbers A000108: (x*c(x))^n + (1/c(x))^n = L(n,-x)= Sum_{k=0..floor(n/2)} T(n,k)*(-x)^k.

This triangle also supplies the absolute values of the coefficients in the multiplication formulas for the Lucas numbers A000032.

From L. Edson Jeffery, Mar 19 2011: (Start)

This sequence is related to rhombus substitution tilings. A signed version of it (see A132460), formed as a triangle with interlaced zeros extending each row to n terms, begins as

{2}

{1, 0}

{1, 0, -2}

{1, 0, -3, 0}

{1, 0, -4, 0, 2}

{1, 0, -5, 0, 5, 0}

....

For the n X n tridiagonal unit-primitive matrix G_(n,1) (n >= 2) (see the L. E. Jeffery link below), defined by

G_(n,1) =

(0 1 0 ... 0)

(1 0 1 0 ... 0)

(0 1 0 1 0 ... 0)

...

(0 ... 0 1 0 1)

(0 ... 0 2 0),

Row n (i.e., {T(n,k)}, k=0..n) of the signed table gives the coefficients of its characteristic function: c_n(x) = Sum_{k=0..n} T(n,k)*x^(n-k) = 0. For example, let n=3. Then

G_(3,1) =

(0 1 0)

(1 0 1)

(0 2 0),

and row 3 of the table is {1,0,-3,0}. Hence c_3(x) = x^3 - 3*x = 0. G_(n,1) has n distinct eigenvalues (the solutions of c_n(x) = 0), given by w_j = 2*cos((2*j-1)*Pi/(2*n)), j=1..n. (End)

For n > 0, T(n,k) is the number of k-subsets of {1,2,...,n} which contain neither consecutive integers nor both 1 and n. Equivalently, T(n,k) is the number of k-subsets without neighbors of a set of n points on a circle. - José H. Nieto S., Jan 17 2012

With the first column omitted, this gives A157000. - Philippe Deléham, Mar 17 2013

The number of necklaces of k black and n - k white beads with no adjacent black beads (Kaplansky 1943). Coefficients of the Dickson polynomials D(n,x,-a). - Peter Bala, Mar 09 2014

From Tom Copeland, Nov 07 2015: (Start)

This triangular array is composed of interleaved rows of reversed, unsigned A127677 (cf. A156308, A217476, A263916) and reversed A111125 (cf. A127672).

See also A113279 for another connection to symmetric and Faber polynomials.

The difference of consecutive rows gives the previous row shifted.

For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 12, 20, and 21) and Damianou and Evripidou (p. 7). (End)

Diagonals are related to multiplicities of eigenvalues of the Laplacian on hyperspheres through A029635. - Tom Copeland, Jan 10 2016

For n>=3, also the independence and matching polynomials of the n-cycle graph C_n. See also A284966. - Eric W. Weisstein, Apr 06 2017

Apparently, with the rows aerated and then the 2s on the diagonal removed, this matrix becomes the reverse, or mirror, of unsigned A117179. See also A114525 - Tom Copeland, May 30 2017

Briggs's (1633) table with an additional column of 2s on the right can be used to generate this table. See p. 69 of the Newton reference. - Tom Copeland, Jun 03 2017

From Liam Solus, Aug 23 2018: (Start)

For n>3 and k>0, T(n,k) equals the number of Markov equivalence classes with skeleton the cycle on n nodes having exactly k immoralities. See Theorem 2.1 of the article by A. Radhakrishnan et al. below.

For n>2 odd and r = floor(n/2)-1, the n-th row is the coefficient vector of the Ehrhart h*-polynomial of the r-stable (n,2)-hypersimplex. See Theorem 4.14 in the article by B. Braun and L. Solus below.

(End)

Conjecture: If a(n) = H(a,b,c,d,n) is a second-order linear recurrence with constant coefficients defined as a(0) = a, a(1)= b, a(n) = c*a(n-1) + d*a(n-2) then a(m*n) = H(a, H(a,b,c,d,m), Sum_{k=0..floor(m/2)} T(m,k)*c^(m-2*k)*d^k, (-1)^(m+1)*d^m, n) (Wolfdieter Lang). - Gary Detlefs, Feb 06 2023

For the proof of the preceding conjecture see the Detlefs and Lang link. There also proofs for several properties of this table are found. - Wolfdieter Lang, Apr 25 2023

From Mohammed Yaseen, Nov 09 2024: (Start)

Let m - 1/m = x, then

m^2 + 1/m^2 = x^2 + 2,

m^3 - 1/m^3 = x^3 + 3*x,

m^4 + 1/m^4 = x^4 + 4*x^2 + 2,

m^5 - 1/m^5 = x^5 + 5*x^3 + 5*x,

m^6 + 1/m^6 = x^6 + 6*x^4 + 9*x^2 + 2,

m^7 - 1/m^7 = x^7 + 7*x^5 + 14*x^3 + 7*x, etc. (End)

N*t^{(N)}n(sigma_1, ..., sigma_N):= sum((x_k)^n, k=1..N) with the elementary symmetric function sigma_k (superscript (N) omitted) in terms of the indeterminates x_1,...,x_N, is an N-variable generalization of Chebyshev's polynomials C_n((sigma_1)/2) = t^{(N=2)}_n(sigma_1, sigma_2 = 1). In general, C_n^{(N)}(sigma_1, ..., sigma{N-1}) := t^{(N)}n(sigma_1, ..., sigma{N-1}, sigma_N:=1). If n > N, one puts sigma_{N+1} = 0, ..., sigma_n = 0.

The sequence of row lengths of this array is A000041(n) (partition numbers).

In row n, this triangular array uses partitions of n listed in the Abramowitz-Stegun order (compare with the M_0, M_1, M_2 and M_3 numbers given in A048996 = |A111786|, A036038, A036039 and A036040, resp.).

Row sums give (-1)^(n-1). Unsigned row sums give A000225(n)= 2^n - 1.

N*t^{(N)}n(sigma_1, ..., sigma_N) gives the sum of the n-th power of the indeterminates x_1, ... , x_N in terms of the elementary symmetric functions of these indeterminates. The coefficient T(n, k) of this partition array corresponds to the k-th partition of n in the Abramowitz-Stegun order, and it multiplies the product of sigma_j functions encoded by this partition. See the example for n = 4 below. - _Wolfdieter Lang, Mar 09 2015

Also equals row squared sums of triangle A132460 and so equals the sum of the initial floor(n/2)+1 squared terms of 1/C(x)^n where C(x) is the g.f. of the Catalan numbers (A000108).

The length of row n is A001400(n), n >= 1.

The Girard-Waring formula for the power sum p(4,n) := Sum_{j=1..4} (x_j)^n in terms of the elementary symmetric functions e_j(x_1, x_2, x_3, x_4), for j = 1, 2, 3, 4, is given by Sum_{i1=0..floor(n/4)} Sum_{i2=0...floor((n-4*i1)/3)} Sum_{i3=0...floor((n-4*i1-3*i2)/2)} ((-1)^(i1 + i3))*n*(n-1-i3-2*i2-3*i1)!/(i1!*i2!*i3!*(n-2*i3-3*i2-4*i1)!)*e_1^(n-2*i3-3*i2-4*i1)*(e_2)^i3*(e_3)^i2*(e_4)^i1, n >= 1 (the arguments of e_j have been omitted). See the W. Lang reference, Theorem 1, case N = 4, with r -> n.

This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions with a part >= 5 eliminated. See row n of the array of Waring numbers A115131, read backwards, with these partitions omitted.

See A135929 and A138034 for further information.

Row sums are A001610(n-1).

Triangle A034807 (coefficients of Lucas polynomials) with the first column omitted. - Philippe Deléham, Mar 17 2013

T(n,k) is the number of ways to select k knights from a round table of n knights, no two adjacent. - Bert Seghers, Mar 02 2014

From Mohammed Yaseen, Nov 09 2024: (Start)

Let m + 1/m = x, then

m^2 + 1/m^2 = x^2 - 2,

m^3 + 1/m^3 = x^3 - 3*x,

m^4 + 1/m^4 = x^4 - 4*x^2 + 2,

m^5 + 1/m^5 = x^5 - 5*x^3 + 5*x,

m^6 + 1/m^6 = x^6 - 6*x^4 + 9*x^2 - 2,

m^7 + 1/m^7 = x^7 - 7*x^5 + 14*x^3 - 7*x, etc. (End)

The length of row n is A001399(n), n >= 1.

The Girard-Waring formula for the power sum p(3,n) = x1^n + x2^2 + x3^n in terms of the elementary symmetric functions e_j(x1, x2, x3), for j=1, 2, 3, is given by Sum_{i=0..floor(n/3)} Sum_{j=0...floor((n-3*i)/2)} ((-1)^j)*n*(n - j - 2*i - 1)!/(i!*j!*(n - 2*j -3*i)!)*e_1^(n-3*i-2*j)*(e_2)^j*(e_3)^i, n >= 1 (the arguments of e_j have been omitted). See the W. Lang reference, Theorem 1, case N = 3, with r -> n.

This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions which have a part larger than 3 elininated. See row n of the array of Waring numbers A115131 read backwards, with these partitions omitted, and numerated with k from 1, 2, ..., A001399(n).

The length of row n is A001401(n), n >= 1.

The Girard-Waring formula for the power sum p(5,n) = Sum_{j=1..5} (x_j)^n in terms of the elementary symmetric functions e_j(x_1, x_2, x_3, x_4), for j = 1, 2 ,..., 5 is given in the W. Lang reference, Theorem 1, in an explicitly nested four sums version. See also the summary link, for N = 5 (there sigma_j^{(N)} -> e_j here).

In this array the partitions of n, with all partitions with a part >= 6 omitted, are used. Here the partitions appear in the reverse Abramowitz-Stegun order. See row n of the array of Waring numbers A115131, read backwards, with the entries corresponding to these omitted partitions.