A193678 -id:A193678 - cp's OEIS frontend

A127672 Monic integer version of Chebyshev T-polynomials (increasing powers).

2, 0, 1, -2, 0, 1, 0, -3, 0, 1, 2, 0, -4, 0, 1, 0, 5, 0, -5, 0, 1, -2, 0, 9, 0, -6, 0, 1, 0, -7, 0, 14, 0, -7, 0, 1, 2, 0, -16, 0, 20, 0, -8, 0, 1, 0, 9, 0, -30, 0, 27, 0, -9, 0, 1, -2, 0, 25, 0, -50, 0, 35, 0, -10, 0, 1, 0, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 2, 0, -36, 0, 105, 0, -112, 0, 54, 0, -12, 0, 1, 0, 13, 0, -91

Offset: 0

Wolfdieter Lang, Mar 07 2007

The row polynomials R(n,x) := Sum_{m=0..n} a(n,m)*x^m have been called Chebyshev C_n(x) polynomials in the Abramowitz-Stegun handbook, p. 778, 22.5.11 (see A049310 for the reference, and note that on p. 774 the S and C polynomials have been mixed up in older printings). - Wolfdieter Lang, Jun 03 2011
This is a signed version of triangle A114525.
The unsigned column sequences (without zeros) are, for m=1..11: A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.
The row polynomials R(n,x) := Sum_{m=0..n} a(n,m)*x*m, give for n=2,3,...,floor(N/2) the positive zeros of the Chebyshev S(N-1,x)-polynomial (see A049310) in terms of its largest zero rho(N):= 2*cos(Pi/N) by putting x=rho(N). The order of the positive zeros is falling: n=1 corresponds to the largest zero rho(N) and n=floor(N/2) to the smallest positive zero. Example N=5: rho(5)=phi (golden section), R(2,phi)= phi^2-2 = phi-1, the second largest (and smallest) positive zero of S(4,x). - Wolfdieter Lang, Dec 01 2010
The row polynomial R(n,x), for n >= 1, factorizes into minimal polynomials of 2*cos(Pi/k), called C(k,x), with coefficients given in A187360, as follows.
  R(n,x) = Product_{d|oddpart(n)} C(2*n/d,x)
         = Product_{d|oddpart(n)} C(2^(k+1)*d,x),
  with oddpart(n)=A000265(n), and 2^k is the largest power of 2 dividing n, where k=0,1,2,...
(Proof: R and C are monic, the degree on both sides coincides, and the zeros of R(n,x) appear all on the r.h.s.) - Wolfdieter Lang, Jul 31 2011 [Theorem 1B, eq. (43) in the W. Lang link. - Wolfdieter Lang, Apr 13 2018]
The zeros of the row polynomials R(n,x) are 2*cos(Pi*(2*k+1)/(2*n)), k=0,1, ..., n-1; n>=1 (from those of the Chebyshev T-polynomials). - Wolfdieter Lang, Sep 17 2011
The discriminants of the row polynomials R(n,x) are found under A193678. - Wolfdieter Lang, Aug 27 2011
The determinant of the N X N matrix M(N) with entries M(N;n,m) = R(m-1,x[n]), 1 <= n,m <= N, N>=1, and any x[n], is identical with twice the Vandermondian Det(V(N)) with matrix entries V(N;n,m) = x[n]^(m-1). This is an instance of the general theorem given in the Vein-Dale reference on p. 59. Note that R(0,x) = 2 (not 1). See also the comments from Aug 26 2013 under A049310 and from Aug 27 2013 under A000178. - Wolfdieter Lang, Aug 27 2013
This triangle a(n,m) is also used to express in the regular (2*(n+1))-gon, inscribed in a circle of radius R, the length ratio side/R, called s(2*(n+1)), as a polynomial in rho(2*(n+1)), the length ratio (smallest diagonal)/side. See the bisections ((-1)^(k-s))*A111125(k,s) and A127677 for comments and examples. - Wolfdieter Lang, Oct 05 2013
From Tom Copeland, Nov 08 2015: (Start)
These are the characteristic polynomials a_n(x) = 2*T_n(x/2) for the adjacency matrix of the Coxeter simple Lie algebra B_n, related to the Cheybshev polynomials of the first kind, T_n(x) = cos(n*q) with x = cos(q) (see p. 20 of Damianou). Given the polynomial (x - t)*(x - 1/t) = 1 - (t + 1/t)*x + x^2 = e2 - e1*x + x^2, the symmetric power sums p_n(t,1/t) = t^n + t^(-n) of the zeros of this polynomial may be expressed in terms of the elementary symmetric polynomials e1 = t + 1/t = y and e2 = t*1/t = 1 as p_n(t,1/t) = a_n(y) = F(n,-y,1,0,0,...), where F(n,b1,b2,...,bn) are the Faber polynomials of A263916.
The partial sum of the first n+1 rows given t and y = t + 1/t is PS(n,t) = Sum_{k=0..n} a_n(y) = (t^(n/2) + t^(-n/2))*(t^((n+1)/2) - t^(-(n+1)/2)) / (t^(1/2) - t^(-1/2)). (For n prime, this is related simply to the cyclotomic polynomials.)
Then a_n(y) = PS(n,t) - PS(n-1,t), and for t = e^(iq), y = 2*cos(q), and, therefore, a_n(2*cos(q)) = PS(n,e^(iq)) - PS(n-1,e^(iq)) = 2*cos(nq) = 2*T_n(cos(q)) with PS(n,e^(iq)) = 2*cos(nq/2)*sin((n+1)q/2) / sin(q/2).
(End)
R(45, x) is the famous polynomial used by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593 to pose four problems, solved by Viète. See, e.g., the Havil reference, pp. 69-74. - Wolfdieter Lang, Apr 28 2018
From Wolfdieter Lang, May 05 2018: (Start)
Some identities for the row polynomials R(n, x) following from the known ones for Chebyshev T-polynomials (A053120) are:
(1) R(-n, x) = R(n, x).
(2) R(n*m, x) = R(n, R(m, x)) = R(m, R(n, x)).
(3) R(2*k+1, x) = (-1)^k*x*S(2*k, sqrt(4-x^2)), k >= 0, with the S row polynomials of A049310.
(4) R(2*k, x) = R(k, x^2-2), k >= 0.
(End)
For y = z^n + z^(-n) and x = z + z^(-1), Hirzebruch notes that y(z) = R(n,x) for the row polynomial of this entry. - Tom Copeland, Nov 09 2019

			Row n=4: [2,0,-4,0,1] stands for the polynomial 2*y^0 - 4*y^2 + 1*y^4. With y^m replaced by 2^(m-1)*x^m this becomes T(4,x) = 1 - 8*x^2 + 8*x^4.
Triangle begins:
n\m   0   1   2   3   4   5   6   7   8   9  10 ...
0:    2
1:    0   1
2:   -2   0   1
3:    0  -3   0   1
4:    2   0  -4   0   1
5:    0   5   0  -5   0   1
6:   -2   0   9   0  -6   0   1
7:    0  -7   0  14   0  -7   0   1
8:    2   0 -16   0  20   0  -8   0   1
9:    0   9   0 -30   0  27   0  -9   0   1
10:  -2   0  25   0 -50   0  35   0 -10   0   1 ...
Factorization into minimal C-polynomials:
R(12,x) = R((2^2)*3,x) = C(24,x)*C(8,x) = C((2^3)*1,x)*C((2^3)*3,x). - _Wolfdieter Lang_, Jul 31 2011

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.
F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 pp. 77, 105.
R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972.
Tom Copeland, Addendum to Elliptic Lie Triad
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2011-2014.
Gary Detlefs and Wolfdieter Lang, Improved Formula for the Multi-Section of the Linear Three-Term Recurrence Sequence, arXiv:2304.12937 [math.CO], 2023.
Wolfdieter Lang, Row polynomials.
Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012-2017.
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.
Index entries for sequences related to Chebyshev polynomials.

Row sums (signed): A057079(n-1). Row sums (unsigned): A000032(n) (Lucas numbers). Alternating row sums: A099837(n+3).
Bisection: A127677 (even n triangle, without zero entries), ((-1)^(n-m))*A111125(n, m) (odd n triangle, without zero entries).
Cf. A049310, A053120, A108045, A263916.

seq(seq(coeff(2*orthopoly[T](n,x/2),x,j),j=0..n),n=0..20); # Robert Israel, Aug 04 2015

a[n_, k_] := SeriesCoefficient[(2 - t*x)/(1 - t*x + x^2), {x, 0, n}, {t, 0, k}]; Flatten[Table[a[n, k], {n, 0, 12}, {k, 0, n}]] (* L. Edson Jeffery, Nov 02 2017 *)

a(n,0) = 0 if n is odd, a(n,0) = 2*(-1)^(n/2) if n is even, else a(n,m) = t(n,m)/2^(m-1) with t(n,m):=A053120(n,m) (coefficients of Chebyshev T-polynomials).
G.f. for m-th column (signed triangle): 2/(1+x^2) if m=0 else (x^m)*(1-x^2)/(1+x^2)^(m+1).
Riordan type matrix ((1-x^2)/(1+x^2),x/(1+x^2)) if one puts a(0,0)=1 (instead of 2).
O.g.f. for row polynomials: R(x,z) := Sum_{n>=0} R(n,x)*z^n = (2-x*z)*S(x,z), with the o.g.f. S(x,z) = 1/(1 - x*z + z^2) for the S-polynomials (see A049310).
Note that R(n,x) = R(2*n,sqrt(2+x)), n>=0 (from the o.g.f.s of both sides). - Wolfdieter Lang, Jun 03 2011
a(n,m) := 0 if n < m or n+m odd; a(n,0) = 2*(-1)^(n/2) (n even); else a(n,m) = ((-1)^((n+m)/2 + m))*n*binomial((n+m)/2-1,m-1)/m.
Recursion for n >= 2 and m >= 2: a(n,m) = a(n-1,m-1) - a(n-2,m), a(n,m) = 0 if n < m, a(2*k,1) = 0, a(2*k+1,1) = (2*k+1)*(-1)^k. In addition, for column m=0: a(2*k,0) = 2*(-1)^k, a(2*k+1,0) = 0, k>=0.
Chebyshev T(n,x) = Sum{m=0..n} a(n,m)*2^(m-1)*x^m. - Wolfdieter Lang, Jun 03 2011
R(n,x) = 2*T(n,x/2) = S(n,x) - S(n-2,x), n>=0, with Chebyshev's T- and S-polynomials, showing that they are integer and monic polynomials. - Wolfdieter Lang, Nov 08 2011
From Tom Copeland, Nov 08 2015: (Start)
a(n,x) = sqrt(2 + a(2n,x)), or 2 + a(2n,x) = a(n,x)^2, is a reflection of the relation of the Chebyshev polynomials of the first kind to the cosine and the half-angle formula, cos(q/2)^2 = (1 + cos(q))/2.
Examples: For n = 2, -2 + x^2 = sqrt(2 + 2 - 4*x^2 + x^4).
For n = 3, -3*x + x^3 = sqrt(2 - 2 + 9*x^2 - 6*x^4 + x^6).
(End)
L(x,h1,h2) = -log(1 - h1*x + h2*x^2) = Sum_{n>0} F(n,-h1,h2,0,...,0) x^n/n = h1*x + (-2*h2 + h1^2) x^2/2 + (-3*h1*h2 + h1^3) x^3/3 + ... is a log series generator of the bivariate row polynomials where T(0,0) = 0 and F(n,b1,b2,...,bn) are the Faber polynomials of A263916. exp(L(x,h1,h2)) = 1 / (1 - h1*x + h2*x^2) is the o.g.f. of A049310. - Tom Copeland, Feb 15 2016

Name changed and table rewritten by Wolfdieter Lang, Nov 08 2011

A127670 Discriminants of Chebyshev S-polynomials A049310.

Original entry on oeis.org

1, 4, 32, 400, 6912, 153664, 4194304, 136048896, 5120000000, 219503494144, 10567230160896, 564668382613504, 33174037869887488, 2125764000000000000, 147573952589676412928, 11034809241396899282944, 884295678882933431599104, 75613185918270483380568064

Offset: 1

Wolfdieter Lang, Jan 23 2007

a(n-1) is the number of fixed n-cell polycubes that are proper in n-1 dimensions (Barequet et al., 2010).
From Rigoberto Florez, Sep 02 2018: (Start)
a(n-1) is the discriminant of the Morgan-Voyce Fibonacci-type polynomial B(n).
Morgan-Voyce Fibonacci-type polynomials are defined as B(0) = 0, B(1) = 1 and B(n) = (x+2)*B(n-1) - B(n-2) for n > 1.
The absolute value of the discriminant of Fibonacci polynomial F(n) is a(n-1).
Fibonacci polynomials are defined as F(0) = 0, F(1) = 1 and F(n) = x*F(n-1) + F(n-2) for n > 1. (End)
The first 6 values are the dimensions of the polynomial ring in 3n variables xi, yi, zi for 1 <= i <= n modulo the ideal generated by x1^a y1^b z1^c + ... + xn^a yn^b zn^c for 0 < a+b+c <= n (see Fact 2.8.1 in Haiman's paper). - Mike Zabrocki, Dec 31 2019

			n=3: The zeros are [sqrt(2),0,-sqrt(2)]. The Vn(xn[1],...,xn[n]) matrix is [[1,1,1],[sqrt(2),0,-sqrt(2)],[2,0,2]]. The squared determinant is 32 = a(3). - _Wolfdieter Lang_, Aug 07 2011

Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf.
G. Barequet and M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes, 31st International Symposium on Computational Geometry (SoCG'15). Editors: Lars Arge and János Pach; pp. 19-22, 2015.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219 for T and U polynomials.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Andrei Asinowski, Gill Barequet, Ronnie Barequet, and Gunter Rote, Proper n-Cell Polycubes in n - 3 Dimensions, Journal of Integer Sequences, Vol. 15 (2012), #12.8.4.
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012. See Th. 1. [From _N. J. A. Sloane_, Oct 16 2010]
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30 (2010), pp. 257-275.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, preprint, 1993.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17--76.
O. Khorunzhiy, Enumeration of tree-type diagrams assembled from oriented chains of edges, arXiv:2207.00766 [math.CO], 2022.
Andrew Snowden, Measures for the colored circle, arXiv:2302.08699 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A007701 (T-polynomials), A086804 (U-polynomials), A171860 and A191092 (fixed n-cell polycubes proper in n-2 and n-3 dimensions, resp.).
Cf. A243953, A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.
A317403 is essentially the same sequence.
Diagonal 1 of A195739.

[((n+1)^n/(n+1)^2)*2^n: n in [1..20]]; // Vincenzo Librandi, Jun 23 2014

Table[((n + 1)^n)/(n + 1)^2 2^n, {n, 1, 30}] (* Vincenzo Librandi, Jun 23 2014 *)

a(n) = ((n+1)^(n-2))*2^n, n >= 1.
a(n) = (Det(Vn(xn[1],...,xn[n])))^2 with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=2*cos(Pi*i/(n+1)), i=1..n, are the zeros of S(n,x):=U(n,x/2).
a(n) = ((-1)^(n*(n-1)/2))*Product_{j=1..n} ((d/dx)S(n,x)|_{x=xn[j]}), n >= 1, with the zeros xn[j], j=1..n, given above.
a(n) = A007830(n-2)*A000079(n), n >= 2. - Omar E. Pol, Aug 27 2011
E.g.f.: -LambertW(-2*x)*(2+LambertW(-2*x))/(4*x). - Vaclav Kotesovec, Jun 22 2014

Slightly edited by Gill Barequet, May 24 2011

A318184 a(n) = 2^(n * (n - 1)/2) * 3^((n - 1) * (n - 2)) * n^(n - 3).

Original entry on oeis.org

1, 1, 72, 186624, 13604889600, 24679069470425088, 1036715783690392172494848, 962459606796748852884396910313472, 19112837387997044228759204010262201783812096, 7926475921550134182551017087135940323782552453120000000, 67406870957147550175650545441605700298239194363455522532832462241792

Offset: 1

Rigoberto Florez, Aug 20 2018

nonn

Discriminant of Fermat polynomials.

F(0)=0, F(1)=1 and F(n) = 3x F(n - 1) -2 F(n - 2) if n>1.

Muniru A Asiru, Table of n, a(n) for n = 1..39
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Fermat Polynomial

Cf. A193678, A007701, A007701, A193678, A303941.

seq(2^(n*(n-1)/2)*3^((n-1)*(n-2))*n^(n-3),n=1..12); # Muniru A Asiru, Dec 07 2018

F[0] = 0; F[1] = 1; F[n_] := F[n] = 3 x F[n - 1] - 2 F[n - 2];
a[n_] := Discriminant[F[n], x];
Array[a, 11] (* Jean-François Alcover, Dec 07 2018 *)

a(n) = 2^(n*(n-1)/2) * 3^((n-1)*(n-2)) * n^(n-3); \\ Michel Marcus, Dec 07 2018

A318197 a(n) = 2^((n - 1)(n + 2)/2)3^(n(n - 1))n^n.

Original entry on oeis.org

1, 144, 629856, 69657034752, 178523361331200000, 10072680467275913619308544, 12094526244510115670028303294529536, 301689370251168256106930569591201258430005248, 153543958878683931150976515367278080485732740052794998784, 1572290138917723454985999517360927544173903258140620787548160000000000

Offset: 1

Rigoberto Florez, Aug 20 2018

nonn

Discriminant of Fermat-Lucas polynomials.

Fermat-Lucas polynomials are defined as F(0) = 2, F(1) = 3*x and F(n) = 3*x*F(n - 1) - 2*F(n - 2) for n > 1.

Andrew Howroyd, Table of n, a(n) for n = 1..30
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
R. Flórez, R. Higuita, and A. Mukherjee, The Star of David and Other Patterns in Hosoya Polynomial Triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Fermat-Lucas polynomials

Cf. A137372, A193678, A007701, A007701, A193678.

[2^((n - 1)*(n + 2) div 2)*3^(n*(n - 1))*n^n: n in [1..10]]; // Vincenzo Librandi, Aug 25 2018

Array[2^((# - 1) (# + 2)/2)*3^(# (# - 1))*#^# &, 10] (* Michael De Vlieger, Aug 22 2018 *)

apply(poldisc, Vec((2-3*x*y)/(1-3*y*x+2*x^2) - 2 + O(x^12))) \\ Andrew Howroyd, Aug 20 2018

a(n) = 2^((n - 1)*(n + 2)/2)*3^(n*(n - 1))*n^n; \\ Andrew Howroyd, Aug 20 2018

A384076 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

1, 1, 153, 2668, 250200, 19423560, 2515242520, 404114856640, 84196030473216, 21703670967664000, 6808856052755927808, 2552126898198385479168, 1126590812208410998119424, 578462173661889165983466496, 341831898528862885226121600000

Offset: 1

Clark Kimberling, May 22 2025

nonn

			The rows of M(4) are (1,3,5,7), (7,1,3,5), (5,7,1,3), (3,5,7,1); determinant(M(4)) = -4716; permanent(M(4)) = 2668, so neg(M(4)) = (-2048 - 7384)/2 = -4716 and pos(M(4)) = (-2048 + 7384)/2 = 2668.

Cf. A193678 (determinant), A384075 (permanent), A380661, A384077, A384078.

z = 19;
v[n_] := Table[2 k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A384074  *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A193678,, with alternating signs *)
neg = (d - p)/2   (* A384075 *)
pos = (d + p)/2   (* A384076 *)

a(n) = (1/2)*(-(-1)^n*A193678(n) + A384074(n)).

A384077 a(n) = neg(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

Original entry on oeis.org

0, -9, -153, -2668, -200200, -20916552, -2515242520, -404114856640, -84096850828032, -21708790967664000, -6808856052755927808, -2552126898198385479168, -1126589571631974396251136, -578462264691449080954733568, -341831898528862885226121600000

Offset: 1

Clark Kimberling, May 29 2025

sign

			The rows of M(4) are (1,3,5,7), (3,5,7,1), (5,7,1,3), (7,1,3,5); determinant(M(4)) = 2048; permanent(M(4)) = 7384, so neg(M(4)) = (7384 - 2048)/2 = -2668 and pos(M(4)) = (7384+2048)/2 = 4716.

Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384078.

z = 15;
v[n_] := Table[2 k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A384074 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A193678 up to signs *)
neg = (d - p)/2    (* A384077 *)
pos = (d + p)/2    (* A384078 *)

a(n) = (1/2)*(s(n)*A193678(n) - A384074(n)), where s(n) = (-1)^((2*n+(-1)^n-1)/4).

A384078 a(n) = pos(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

1, 1, 45, 4716, 250200, 19423560, 2462535768, 406262340288, 84196030473216, 21703670967664000, 6808563893605222144, 2552145158372103507456, 1126590812208410998119424, 578462173661889165983466496, 341831891354409385226121600000

Offset: 1

Clark Kimberling, Jun 01 2025

nonn

			The rows of M(4) are (1,3,5,7), (3,5,7,1), (5,7,1,3), (7,1,3,5); determinant(M(4)) = 2048; permanent(M(4)) = 7384, so neg(M(4)) = (7384 - 2048)/2 = -2668 and pos(M(4)) = (7384+2048)/2 = 4716.

Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384077.

z = 15;
v[n_] := Table[2 k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]     (* A384074 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A193678 up to signs *)
neg = (d - p)/2    (* A384077 *)
pos = (d + p)/2    (* A384078 *)

a(n) = (1/2)*(s(n)*A193678(n) - A384074(n)), where s(n) = (-1)^((2*n+(-1)^n-1)/4).

A384074 a(n) = permanent of the n X n circulant matrix with (row 1) = (1, 3, 5, 7, ..., 2n - 1).

Original entry on oeis.org

1, 10, 198, 7384, 450400, 40340112, 4977778288, 810377196928, 168292881301248, 43412461935328000, 13617419946361149952, 5104272056570488986624, 2253180383840385394370560, 1156924438353338246938200064, 683663789883272270452243200000

Offset: 1

Clark Kimberling, May 22 2025

nonn

Cf. A005408, A193678 (determinant), A384075, A384076, A384077, A384078.

z = 16; v[n_] := Table[2 k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]];
Table[Simplify[Permanent[u[n]]], {n, 1, z}]

A086804 a(0)=0; for n > 0, a(n) = (n+1)^(n-2)*2^(n^2).

Original entry on oeis.org

0, 1, 16, 2048, 1638400, 7247757312, 164995463643136, 18446744073709551616, 9803356117276277820358656, 24178516392292583494123520000000, 271732164163901599116133024293512544256

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

Discriminant of Chebyshev polynomial U_n (x) of second kind.
Chebyshev second kind polynomials are defined by U(0)=0, U(1)=1 and U(n) = 2xU(n-1) - U(n-2) for n > 1.
The absolute value of the discriminant of Pell polynomials is a(n-1).
Pell polynomials are defined by P(0)=0, P(1)=1 and P(n) = 2x P(n-1) + P(n-2) if n > 1. - Rigoberto Florez, Sep 01 2018

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219, 5.1.2.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
Eric Weisstein's World of Mathematics, Pell Polynomial
Index entries for sequences related to Chebyshev polynomials.

Cf. A007701, A127670 (discriminant for S-polynomials), A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

[0] cat [(n+1)^(n-2)*2^(n^2): n in [1..10]]; // G. C. Greubel, Nov 11 2018

Join[{0},Table[(n+1)^(n-2) 2^n^2,{n,10}]] (* Harvey P. Dale, May 01 2015 *)

PARI
```
a(n)=if(n<1,0,(n+1)^(n-2)*2^(n^2))
```

a(n)=if(n<1,0,n++; poldisc(poltchebi(n)'/n))

a(n) = ((n+1)^(n-2))*2^(n^2), n >= 1, a(0):=0.
a(n) = ((2^(2*(n-1)))*Det(Vn(xn[1],...,xn[n])))^2, n >= 1, with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=cos(Pi*i/(n+1)), i=1..n, are the zeros of the Chebyshev U(n,x) polynomials.
a(n) = ((-1)^(n*(n-1)/2))*(2^(n*(n-2)))*Product_{i=1..n}((d/dx)U(n,x)|_{x=xn[i]}), n >= 1, with the zeros xn[i], i=1..n, given above.

Formula and more terms from Vladeta Jovovic, Aug 07 2003

cp's OEIS Frontend

A127672 Monic integer version of Chebyshev T-polynomials (increasing powers).

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A127670 Discriminants of Chebyshev S-polynomials A049310.

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A318184 a(n) = 2^(n * (n - 1)/2) * 3^((n - 1) * (n - 2)) * n^(n - 3).

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A318197 a(n) = 2^((n - 1)*(n + 2)/2)*3^(n*(n - 1))*n^n.

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A384076 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

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A384077 a(n) = neg(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

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A384078 a(n) = pos(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

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A384074 a(n) = permanent of the n X n circulant matrix with (row 1) = (1, 3, 5, 7, ..., 2n - 1).

A318197 a(n) = 2^((n - 1)(n + 2)/2)3^(n(n - 1))n^n.