A127672 -id:A127672 - cp's OEIS frontend

A034807 Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.

2, 1, 1, 2, 1, 3, 1, 4, 2, 1, 5, 5, 1, 6, 9, 2, 1, 7, 14, 7, 1, 8, 20, 16, 2, 1, 9, 27, 30, 9, 1, 10, 35, 50, 25, 2, 1, 11, 44, 77, 55, 11, 1, 12, 54, 112, 105, 36, 2, 1, 13, 65, 156, 182, 91, 13, 1, 14, 77, 210, 294, 196, 49, 2, 1, 15, 90, 275, 450, 378, 140, 15, 1, 16, 104

Offset: 0

N. J. A. Sloane

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)

			I have seen two versions of these polynomials: One version begins L_0 = 2, L_1 = 1, L_2 = 1 + 2*x, L_3 = 1 + 3*x, L_4 = 1 + 4*x + 2*x^2, L_5 = 1 + 5*x + 5*x^2, L_6 = 1 + 6*x + 9*x^2 + 2*x^3, L_7 = 1 + 7*x + 14*x^2 + 7*x^3, L_8 = 1 + 8*x + 20*x^2 + 16*x^3 + 2*x^4, L_9 = 1 + 9*x + 27*x^2 + 30*x^3 + 9*x^4, ...
The other version (probably the more official one) begins L_0(x) = 2, L_1(x) = x, L_2(x) = 2 + x^2, L_3(x) = 3*x + x^3, L_4(x) = 2 + 4*x^2 + x^4, L_5(x) = 5*x + 5*x^3 + x^5, L_6(x) = 2 + 9*x^2 + 6*x^4 + x^6, L_7(x) = 7*x + 14*x^3 + 7*x^5 + x^7, L_8(x) = 2 + 16*x^2 + 20*x^4 + 8*x^6 + x^8, L_9(x) = 9*x + 30*x^3 + 27*x^5 + 9*x^7 + x^9.
From _John Blythe Dobson_, Oct 11 2007: (Start)
Triangle begins:
  2;
  1;
  1,  2;
  1,  3;
  1,  4,  2;
  1,  5,  5;
  1,  6,  9,   2;
  1,  7, 14,   7;
  1,  8, 20,  16,   2;
  1,  9, 27,  30,   9;
  1, 10, 35,  50,  25,   2;
  1, 11, 44,  77,  55,  11;
  1, 12, 54, 112, 105,  36,   2;
  1, 13, 65, 156, 182,  91,  13;
  1, 14, 77, 210, 294, 196,  49,  2;
  1, 15, 90, 275, 450, 378, 140, 15;
(End)
From _Peter Bala_, Mar 20 2025: (Start)
Let S = x + y and M = -x*y. Then the triangle gives the coefficients when expressing the symmetric polynomial x^n + y^n as a polynomial in S and M. For example,
x^2 + y^2 = S^2 + 2*M; x^3 + y^3 = S^3 + 3*S*M; x^4 + y^4 = S^4 + 4*(S^2)*M + 2*M^2;
x^5 + y^5 = S^5 + 5*(S^3)*M + 5*S*M^2; x^6 + y^6 = S^6 + 6*(S^4)*M + 9*(S^2)*M^2 + 2*M^3. See Woko. In general x^n + y^n = 2*(-i)^n *(sqrt(M))^n * T(n, i*S/(2*sqrt(M))), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. (End)

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 148.
C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
Thomas Koshy, Fibonacci and Lucas Numbers with Applications. New York, etc.: John Wiley & Sons, 2001. (Chapter 13, "Pascal-like Triangles," is devoted to the present triangle.)
The Royal Society Newton Tercentenary Celebrations, Cambridge Univ. Press, 1947.

T. D. Noe, Rows n = 0..100 of triangle, flattened
Moussa Benoumhani, A Sequence of Binomial Coefficients Related to Lucas and Fibonacci Numbers, J. Integer Seqs., Vol. 6, 2003.
Benjamin Braun and Liam Solus, r-stable hypersimplices, arXiv:1408.4713 [math.CO], 2014-2016; Journal of Combinatorial Theory, Series A 157 (2018): 349-388.
Johann Cigler and Hans-Christian Herbig, Factorization of spread polynomials, arXiv:2412.18958 [math.NT], 2024. See p. 2.
Tom Copeland, Addendum to Elliptic Lie Triad
Pantelis A. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Pantelis A. Damianou and Charalampos A. Evripidou, Characteristic and Coxeter polynomials for affine Lie algebras, arXiv preprint arXiv:1409.3956 [math.RT], 2014.
Gary Detlefs and Wolfdieter Lang, Improved Formula for the Multi-Section of the Linear Three-Term Rcurrence Sequence<, arXiv:2304.12937[nath.CO], 2023.
Joseph Doolittle, Lukas Katthän, Benjamin Nill, and Francisco Santos, Empty simplices of large width, arXiv:2103.14925 [math.CO], 2021.
Sergio Falcon, On the Lucas triangle and its relationship with the k-Lucas numbers, Journal of Mathematical and Computational Science, 2 (2012), No. 3, 425-434.
E. J. Farrell, An introduction to matching polynomials, J. Combin. Theory B 27 (1) (1979), 75-86, Table 2.
Heidi Goodson, An Identity for Vertically Aligned Entries in Pascal's Triangle, arXiv:1901.08653 [math.CO], 2019.
Gábor Hetyei, Hurwitzian continued fractions containing a repeated constant and an arithmetic progression, arXiv preprint arXiv:1211.2494 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 02 2013
L. E. Jeffery, Unit-primitive matrices
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Feihu Liu, Ying Wang, Yingrui Zhang, and Zihao Zhang,Hankel Determinants for Convolution of Power Series: An Extension of Cigler's Results, arXiv:2503.17187 [math.CO], 2025. See p.5.
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Eric Weisstein's World of Mathematics, Cycle Graph
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Wikipedia, Metallic mean.
Justin Eze Woko, A Pascal-like Triangle for alpha^n + beta^n, The Mathematical Gazette, Vol. 81, No. 490 (Mar., 1997), pp. 75-79.

Cf. A029635, A061896, A111125, A113279, A114525, A127672, A127677, A156308, A217476, A263916.

Cf. A117179.

T:= proc(n,k) if n=0 and k=0 then 2 elif k>floor(n/2) then 0 else n*binomial(n-k,k)/(n-k) fi end: for n from 0 to 15 do seq(T(n,k), k=0..floor(n/2)) od; # yields sequence in triangular form # Emeric Deutsch, Dec 25 2004

t[0, 0] = 2; t[n_, k_] := Binomial[n-k, k] + Binomial[n-k-1, k-1]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Dec 30 2013 *)
CoefficientList[Table[x^(n/2) LucasL[n, 1/Sqrt[x]], {n, 0, 15}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
Table[Select[Reverse[CoefficientList[LucasL[n, x], x]], 0 < # &], {n, 0, 16}] // Flatten (* Robert G. Wilson v, May 03 2017 *)
CoefficientList[FunctionExpand @ Table[2 (-x)^(n/2) Cos[n ArcSec[2 Sqrt[-x]]], {n, 0, 15}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
CoefficientList[Table[2 (-x)^(n/2) ChebyshevT[n, 1/(2 Sqrt[-x])], {n, 0, 15}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)

{T(n, k) = if( k<0 || 2*k>n, 0, binomial(n-k, k) + binomial(n-k-1, k-1) + (n==0))}; /* Michael Somos, Jul 15 2003 */

Row sums = A000032. T(2n, n-1) = A000290(n), T(2n+1, n-1) = A000330(n), T(2n, n-2) = A002415(n). T(n, k) = A029635(n-k, k), if n>0. - Michael Somos, Apr 02 1999
Lucas polynomial coefficients: 1, -n, n*(n-3)/2!, -n*(n-4)*(n-5)/3!, n*(n-5)*(n-6)*(n-7)/4!, - n*(n-6)*(n-7)*(n-8)*(n-9)/5!, ... - Herb Conn and Gary W. Adamson, May 28 2003
G.f.: (2-x)/(1-x-x^2*y). - Vladeta Jovovic, May 31 2003
T(n, k) = T(n-1, k) + T(n-2, k-1), n>1. T(n, 0) = 1, n>0. T(n, k) = binomial(n-k, k) + binomial(n-k-1, k-1) = n*binomial(n-k-1, k-1)/k, 0 <= 2*k <= n except T(0, 0) = 2. - Michael Somos, Apr 02 1999
T(n,k) = (n*(n-1-k)!)/(k!*(n-2*k)!), n>0, k>=0. - Alexander Elkins (alexander_elkins(AT)hotmail.com), Jun 09 2007
O.g.f.: 2-(2xt+1)xt/(-t+xt+(xt)^2). (Cf. A113279.) - Tom Copeland, Nov 07 2015
T(n,k) = A011973(n-1,k) + A011973(n-3,k-1) = A011973(n,k) - A011973(n-4,k-2) except for T(0,0)=T(2,1)=2. - Xiangyu Chen, Dec 24 2020
L_n(x) = ((x+sqrt(x^2+4))/2)^n + (-((x+sqrt(x^2+4))/2))^(-n). See metallic means. - William Krier, Sep 01 2023

Improved description, more terms, etc., from Michael Somos

A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.

Original entry on oeis.org

1, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1

Offset: 0

Paul Barry, Oct 27 2004

A transform of (-1)^n.
Row sums of Riordan array ((1-x)/(1+x), x/(1+x)^2), A110162.
Let b(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)(-1)^(n-2k). Then a(n) = b(n) - b(n-2) = A049347(n) - A049347(n-2) (n > 0). The g.f. 1/(1+x) of (-1)^n is transformed to (1-x^2)/(1+x+x^2) under the mapping G(x)->((1-x^2)/(1+x^2))G(x/(1+x^2)). Partial sums of A099838.
A(n) = a(n+3) (or a(n) if a(0) is replaced by 2) appears, together with B(n) = A049347(n) in the formula 2*exp(2*Pi*n*i/3) = A(n) + B(n)*sqrt(3)*i, n >= 0, with i = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014

			G.f. = 1 - x - x^2 + 2*x^3 - x^4 - x^5 + 2*x^6 - x^7 - x^8 + 2*x^9 - x^10 + ...

Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Cf. A061347, A100051, A100063, A098554, A110162.

A099837 := proc(n)
    option remember;
    if n <=2 then
        op(n+1,[1,-1,-1]) ;
    else
        -procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A099837(n),n=0..80) ; # R. J. Mathar, Apr 26 2022

a[0] = 1; a[n_] := Mod[n+2, 3] - Mod[n, 3]; A099837 = Table[a[n], {n, 0, 71}](* Jean-François Alcover, Feb 15 2012, after Michael Somos *)
LinearRecurrence[{-1, -1}, {1, -1, -1}, 50] (* G. C. Greubel, Aug 08 2017 *)

A099837(n) := block(
        if n = 0 then 1 else [2,-1,-1][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

{a(n) = [2, -1, -1][n%3 + 1] - (n == 0)}; /* Michael Somos, Jan 19 2012 */

Vec((1-x^2)/(1+x+x^2) + O(x^20)) \\ Felix Fröhlich, Aug 08 2017

G.f.: (1-x^2)/(1+x+x^2).
Euler transform of length 3 sequence [-1, -1, 1]. - Michael Somos, Mar 21 2011
Moebius transform is length 3 sequence [-1, 0, 3]. - Michael Somos, Mar 22 2011
a(n) = -b(n) where b(n) = A061347(n) is multiplicative with b(3^e) = -2 if e > 0, b(p^e) = 1 otherwise. - Michael Somos, Jan 19 2012
a(n) = a(-n). a(n) = c_3(n) if n > 1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
G.f.: (1 - x) * (1 - x^2) / (1 - x^3). a(n) = -a(n-1) - a(n-2) unless n = 0, 1, 2. - Michael Somos, Jan 19 2012
Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)*(3^(1-s)-1). - R. J. Mathar, Apr 11 2011
a(n+3) = R(n,-1) for n >= 0, with the monic Chebyshev T-polynomials R with coefficient table A127672. - Wolfdieter Lang, Feb 27 2014
For n > 0, a(n) = 2*cos(n*Pi/3)*cos(n*Pi). - Wesley Ivan Hurt, Sep 25 2017
From Peter Bala, Apr 20 2024: (Start)
a(n) is equal to the n-th order Taylor polynomial (centered at 0) of 1/c(x)^(2*n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. Cf. A333093.
Row sums of the Riordan array A110162. (End)

A005581 a(n) = (n-1)n(n+4)/6.

Original entry on oeis.org

0, 0, 2, 7, 16, 30, 50, 77, 112, 156, 210, 275, 352, 442, 546, 665, 800, 952, 1122, 1311, 1520, 1750, 2002, 2277, 2576, 2900, 3250, 3627, 4032, 4466, 4930, 5425, 5952, 6512, 7106, 7735, 8400, 9102, 9842, 10621, 11440, 12300, 13202, 14147, 15136, 16170

Offset: 0

N. J. A. Sloane

A class of Boolean functions of n variables and rank 2.
Also, number of inscribable triangles within a (n+4)-gon sharing with them its vertices but not its sides. - Lekraj Beedassy, Nov 14 2003
a(n) = A111808(n,3) for n > 2. - Reinhard Zumkeller, Aug 17 2005
If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to the number of (n-3)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
The sequence starting with offset 2 = binomial transform of [2, 5, 4, 1, 0, 0, 0, ...]. - Gary W. Adamson, Mar 20 2009
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 4, a(n-4) is the number of (0,1) n X n matrices A <= P^(-1) + I + P having exactly two 1's in every row and column with perA=8. - Vladimir Shevelev, Apr 12 2010
Also arises as the number of triples of edges which can be chosen as the cut-points in the "three-opt" heuristic for a traveling salesman problem on (n+4) nodes. - James McDermott, Jul 10 2015
a(n) = risefac(n, 3)/3! - n is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 3 and dimension n. Here risefac is the rising factorial. - Wolfdieter Lang, Dec 10 2015
For n >= 2, a(n) is the number of characters in a word Q formed by concatenating all 'directed' ( left to right or vice versa), unrearranged subwords, from length 1 to (n-1), of a length (n-1) word q- allowing for the appearance of repeated subwords- and simply inserting an extra character for all subwords thus concatenated. - Christopher Hohl, May 30 2019

			In hexagon ABCDEF, the "interior" triangles are ACE and BDF, and a(6-4)=a(2)=2. - _Toby Gottfried_, Nov 12 2011
G.f. = 2*x^2 + 7*x^3 + 16*x^4 + 30*x^5 + 50*x^6 + 77*x^7 + 112*x^8 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
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Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Dead link]
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Cached copy, May 15 2013]
Eric Weisstein's World of Mathematics, Trinomial Coefficient.
Index entries for sequences related to Boolean functions.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000217, A000292, A005582, A005586, A111808.
Cf. A176222, A000211, A052928, A128209.
Cf. A127672 (Chebyshev C).

[(n-1)*n*(n+4)/6 : n in [0..50]]; // Wesley Ivan Hurt, Jul 10 2015

A005581 := n->(n-1)*n*(n+4)/6: seq(A005581(n), n=0..50);
a:=n->sum ((j+3)*j/2,j=0..n): seq(a(n),n=-1..49); # Zerinvary Lajos, Dec 17 2006
seq((n+3)*binomial(n,3)/n, n=1..46); # Zerinvary Lajos, Feb 28 2007
A005581:=-(-2+z)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
seq(sum(binomial(n,m), m=1..3)+n^2,n=-1..44); # Zerinvary Lajos, Jun 19 2008
A005581 := n -> GegenbauerC(`if`(3A005581(n)), n=0..50); # Peter Luschny, May 10 2016

Table[(n-1)*n*(n+4)/6, {n,0,50}] (* Stefan Steinerberger, Apr 10 2006 *)
LinearRecurrence[{4,-6,4,-1},{0,0,2,7},50] (* Harvey P. Dale, Sep 22 2012 *)

A005581(n):=(n-1)*n*(n+4)/6$ makelist(A005581(n),n,0,50); /* Martin Ettl, Dec 18 2012 */

{a(n) = n * (n+4) * (n-1) / 6}; /* Michael Somos, Apr 13 2007 */

concat([0, 0], Vec((x^2)*(2-x)/(1-x)^4 + O(x^50))) \\ Altug Alkan, Dec 10 2015

[(n-1)*n*(n+4)/6 for n in range(50)] # Danny Rorabaugh, Apr 20 2015

G.f.: (x^2)*(2-x)/(1-x)^4.
a(n) = binomial(n+1, n-2) + binomial(n, n-2).
a(n) = A027907(n, 3), n >= 0 (fourth column of trinomial coefficients). - N. J. A. Sloane, May 16 2003
Convolution of {1, 2, 3, ...} with {2, 3, 4, ...}. - Jon Perry, Jun 25 2003
a(n+2) = 2*te(n) - te(n-1), e.g., a(5) = 2*te(3) - te(2) = 2*20 - 10 = 30, where te(n) are the tetrahedral numbers A000292. - Jon Perry, Jul 23 2003
a(n) is the coefficient of x^3 in the expansion of (1+x+x^2)^n. For example, a(1)=0 since (1+x+x^2)^1=1+x+x^2. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 09 2007
E.g.f.: (x^2 + x^3/6) * exp(x). - Michael Somos, Apr 13 2007
a(n) = - A005586(-4-n) for all n in Z. - Michael Somos, Apr 13 2007
a(n) = C(4+n,3)-(n+4)*(n+1), since C(4+n,3) = number of all triangles in (n+4)-gon, and (n+4)*(n+1)=number of triangles with at least one of the edges included. Example: n=0,in a square, all 4 possible triangles include some of the square's edges and C(4+n,3)-(n+4)*(n+1)=4-4*1=0 = number of other triangles = a(0). - Toby Gottfried, Nov 12 2011
a(n) = 2*binomial(n,2) + binomial(n,3). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(0)=0, a(1)=0, a(2)=2, a(3)=7, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Sep 22 2012
a(n) = A000292(n-1) + A000217(n-1) for all n in Z. - Michael Somos, Jul 29 2015
a(n+2) = -A127672(6+n, n), n >= 0, with A127672 giving the coefficients of Chebyshev's C polynomials. See the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 10 2015
a(n) = GegenbauerC(N, -n, -1/2) where N = 3 if 3Peter Luschny, May 10 2016
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=2} 1/a(n) = 163/200.
Sum_{n>=2} (-1)^n/a(n) = 12*log(2)/5 - 253/200. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000

A111125 Triangle read by rows: T(k,s) = ((2k+1)/(2s+1))binomial(k+s,2s), 0 <= s <= k.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 7, 14, 7, 1, 9, 30, 27, 9, 1, 11, 55, 77, 44, 11, 1, 13, 91, 182, 156, 65, 13, 1, 15, 140, 378, 450, 275, 90, 15, 1, 17, 204, 714, 1122, 935, 442, 119, 17, 1, 19, 285, 1254, 2508, 2717, 1729, 665, 152, 19, 1, 21, 385, 2079, 5148, 7007, 5733, 2940, 952, 189, 21, 1

Offset: 0

N. J. A. Sloane, Oct 16 2005

Riordan array ((1+x)/(1-x)^2, x/(1-x)^2). Row sums are A002878. Diagonal sums are A003945. Inverse is A113187. An interesting factorization is (1/(1-x), x/(1-x))(1+2*x, x*(1+x)). - Paul Barry, Oct 17 2005
Central coefficients of rows with odd numbers of term are A052227.
From Wolfdieter Lang, Jun 26 2011: (Start)
T(k,s) appears as T_s(k) in the Knuth reference, p. 285.
This triangle is related to triangle A156308(n,m), appearing in this reference as U_m(n) on p. 285, by T(k,s) - T(k-1,s) = A156308(k,s), k>=s>=1 (identity on p. 286). T(k,s) = A156308(k+1,s+1) - A156308(k,s+1), k>=s>=0 (identity on p. 286).
(End)
A111125 is jointly generated with A208513 as an array of coefficients of polynomials v(n,x): initially, u(1,x)= v(1,x)= 1; for n>1, u(n,x)= u(n-1,x) +x*(x+1)*v(n-1) and v(n,x)= u(n-1,x) +x*v(n-1,x) +1. See the Mathematica section. The columns of A111125 are identical to those of A208508. Here, however, the alternating row sums are periodic (with period 1,2,1,-1,-2,-1). - Clark Kimberling, Feb 28 2012
This triangle T(k,s) (with signs and columns scaled with powers of 5) appears in the expansion of Fibonacci numbers F=A000045 with multiples of odd numbers as indices in terms of odd powers of F-numbers. See the Jennings reference, p. 108, Theorem 1. Quoted as Lemma 3 in the Ozeki reference given in A111418. The formula is: F_{(2*k+1)*n} = Sum_{s=0..k} ( T(k,s)*(-1)^((k+s)*n)*5^s*F_{n}^(2*s+1) ), k >= 0, n >= 0. - Wolfdieter Lang, Aug 24 2012
From Wolfdieter Lang, Oct 18 2012: (Start)
This triangle T(k,s) appears in the formula x^(2*k+1) - x^(-(2*k+1)) = Sum_{s=0..k} ( T(k,s)*(x-x^(-1))^(2*s+1) ), k>=0. Prove the inverse formula (due to the Riordan property this will suffice) with the binomial theorem. Motivated to look into this by the quoted paper of Wang and Zhang, eq. (1.4).
Alternating row sums are A057079.
The Z-sequence of this Riordan array is A217477, and the A-sequence is (-1)^n*A115141(n). For the notion of A- and Z-sequences for Riordan triangles see a W. Lang link under A006232. (End)
The signed triangle ((-1)^(k-s))*T(k,s) gives the coefficients of (x^2)^s of the polynomials C(2*k+1,x)/x, with C the monic integer Chebyshev T-polynomials whose coefficients are given in A127672 (C is there called R). See the odd numbered rows there. This signed triangle is the Riordan array ((1-x)/(1+x)^2, x/(1+x)^2). Proof by comparing the o.g.f. of the row polynomials where x is replaced by x^2 with the odd part of the bisection of the o.g.f. for C(n,x)/x. - Wolfdieter Lang, Oct 23 2012
From Wolfdieter Lang, Oct 04 2013: (Start)
The signed triangle S(k,s) := ((-1)^(k-s))*T(k,s) (see the preceding comment) is used to express in a (4*(k+1))-gon the length ratio s(4*(k+1)) = 2*sin(Pi/4*(k+1)) =  2*cos((2*k+1)*Pi/(4*(k+1))) of a side/radius as a polynomial in rho(4*(k+1)) = 2*cos(Pi/4*(k+1)), the length ratio (smallest diagonal)/side:
  s(4*(k+1)) = Sum_{s=0..k} ( S(k,s)*rho(4*(k+1))^(2*s+1) ).
This is to be computed modulo C(4*(k+1), rho(4*(k+1)) = 0, the minimal polynomial (see A187360) in order to obtain  s(4*(k+1)) as an integer in the algebraic number field Q(rho(4*(k+1))) of degree delta(4*(k+1)) (see A055034). Thanks go to Seppo Mustonen for asking me to look into the problem of the square of the total length in a regular n-gon, where this formula is used in the even n case. See A127677 for the formula in the (4*k+2)-gon. (End)
From Wolfdieter Lang, Aug 14 2014: (Start)
The row polynomials for the signed triangle (see the Oct 23 2012 comment above), call them todd(k,x) = Sum_{s=0..k} ( (-1)^(k-s)*T(k,s)*x^s ) = S(k, x-2) - S(k-1, x-2), k >= 0, with the Chebyshev S-polynomials (see their coefficient triangle (A049310) and S(-1, x) = 0), satisfy the recurrence todd(k, x) = (-1)^(k-1)*((x-4)/2)*todd(k-1, 4-x) + ((x-2)/2)*todd(k-1, x), k >= 1, todd(0, x) = 1. From the Aug 03 2014 comment on A130777.
This leads to a recurrence for the signed triangle, call it S(k,s) as in the Oct 04 2013 comment: S(k,s) = (1/2)*(1 + (-1)^(k-s))*S(k-1,s-1) + (2*(s+1)*(-1)^(k-s) - 1)*S(k-1,s) + (1/2)*(-1)^(k-s)*Sum_{j=0..k-s-2} ( binomial(j+s+2,s)*4^(j+2)* S(k-1, s+1+j) ) for k >= s >= 1, and S(k,s) = 0 if k < s and S(k,0) = (-1)^k*(2*k+1). Note that the recurrence derived from the Riordan A-sequence A115141 is similar but has simpler coefficients: S(k,s) = sum(A115141(j)*S(k-1,s-1+j), j=0..k-s), k >= s >=1.
(End)
From Tom Copeland, Nov 07 2015: (Start)
Rephrasing notes here: Append an initial column of zeros, except for a 1 at the top, to A111125 here. Then the partial sums of the columns of this modified entry are contained in A208513. Append an initial row of zeros to A208513. Then the difference of consecutive pairs of rows of the modified A208513 generates the modified A111125. Cf. A034807 and A127677.
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. 20 and 21) and Damianou and Evripidou (p. 7).
As suggested by the equations on p. 7 of Damianou and Evripidou, the signed row polynomials of this entry are given by (p(n,x))^2 = (A(2*n+1, x) + 2)/x = (F(2*n+1, (2-x), 1, 0, 0, ... ) + 2)/x = F(2*n+1, -x, 2*x, -3*x, ..., (-1)^n n*x)/x = -F(2*n+1, x, 2*x, 3*x, ..., n*x)/x, where A(n,x) are the polynomials of A127677 and F(n, ...) are the Faber polynomials of A263196. Cf. A127672 and A127677.
(End)
The row polynomials P(k, x) of the signed triangle S(k, s) = ((-1)^(k-s))*T(k, s) are given from the row polynomials R(2*k+1, x) of triangle A127672 by
P(k, x) =  R(2*k+1, sqrt(x))/sqrt(x). - Wolfdieter Lang, May 02 2021

			Triangle T(k,s) begins:
k\s  0    1     2     3     4     5     6    7    8   9 10
0:   1
1:   3    1
2:   5    5     1
3:   7   14     7     1
4:   9   30    27     9     1
5:  11   55    77    44    11     1
6:  13   91   182   156    65    13     1
7:  15  140   378   450   275    90    15    1
8:  17  204   714  1122   935   442   119   17    1
9:  19  285  1254  2508  2717  1729   665  152   19   1
10: 21  385  2079  5148  7007  5733  2940  952  189  21  1
... Extended and reformatted by _Wolfdieter Lang_, Oct 18 2012
Application for Fibonacci numbers F_{(2*k+1)*n}, row k=3:
F_{7*n} = 7*(-1)^(3*n)*F_n + 14*(-1)^(4*n)*5*F_n^3 + 7*(-1)^(5*n)*5^2*F_n^5 + 1*(-1)^(6*n)*5^3*F_n^7, n>=0. - _Wolfdieter Lang_, Aug 24 2012
Example for the  Z- and A-sequence recurrences  of this Riordan triangle: Z = A217477 = [3,-4,12,-40,...]; T(4,0) = 3*7 -4*14 +12*7 -40*1 = 9. A =  [1, 2, -1, 2, -5, 14, ..]; T(5,2) = 1*30 + 2*27 - 1*9 + 2*1= 77. _Wolfdieter Lang_, Oct 18 2012
Example for the (4*(k+1))-gon length ratio s(4*(k+1))(side/radius) as polynomial in the ratio rho(4*(k+1)) ((smallest diagonal)/side): k=0, s(4) = 1*rho(4) = sqrt(2); k=1, s(8) = -3*rho(8) + rho(8)^3 = sqrt(2-sqrt(2)); k=2, s(12) = 5*rho(12) - 5*rho(12)^3 + rho(12)^5, and C(12,x) = x^4 - 4*x^2 + 1, hence rho(12)^5 = 4*rho(12)^3 - rho(12), and s(12) = 4*rho(12) - rho(12)^3 = sqrt(2 - sqrt(3)). - _Wolfdieter Lang_, Oct 04 2013
Example for the recurrence for the signed triangle S(k,s)= ((-1)^(k-s))*T(k,s) (see the Aug 14 2014 comment above):
S(4,1) = 0 + (-2*2 - 1)*S(3,1) - (1/2)*(3*4^2*S(3,2) + 4*4^3*S(3,3)) = - 5*14 - 3*8*(-7) - 128*1 = -30. The recurrence from the Riordan A-sequence A115141 is S(4,1) = -7 -2*14 -(-7) -2*1 = -30. - _Wolfdieter Lang_, Aug 14 2014

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
R. Andre-Jeannin, A generalization of Morgan-Voyce polynomials, The Fibonacci Quarterly 32.3 (1994): 228-31.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681.
P. Damianou , On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
P. Damianou and C. Evripidou, Characteristic and Coxeter polynomials for affine Lie algebras, arXiv preprint arXiv:1409.3956 [math.RT], 2014.
D. Jennings, Some Polynomial Identities for the Fibonacci and Lucas Numbers, Fib. Quart., 31(2) (1993), 134-137.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), no. 203, 277-294.
Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart., Nov. 2005, pp. 359-370.
T. Wang and W. Zhang, Some identities involving Fibonacci, Lucas polynomials and their applications, Bull. Math. Soc. Sci. Math. Roumanie, Tome 55(103), No.1, (2012) 95-103.
Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials

Mirror image of A082985, which see for further references, etc.
Also closely related to triangles in A098599 and A100218.
Cf. A052227, A085478, A208513, A208508, A211957
Cf. A034807, A127672, A127677, A263196.

[((2*n+1)/(n+k+1))*Binomial(n+k+1, 2*k+1): k in [0..n], n in [0..12]];  // G. C. Greubel, Feb 01 2022

(* First program *)
u[1, x_]:=1; v[1, x_]:=1; z=16;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A208513 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A111125 *) (* Clark Kimberling, Feb 28 2012 *)
(* Second program *)
T[n_, k_]:= ((2*n+1)/(2*k+1))*Binomial[n+k, 2*k];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 01 2022 *)

@CachedFunction
def T(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 3*T(n-1,k) if n==1 else 2*T(n-1,k)
    return T(n-1,k-1) - T(n-2,k) - h
A111125 = lambda n,k: (-1)^(n-k)*T(n,k)
for n in (0..9): [A111125(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
From Peter Bala, Apr 30 2012: (Start)
T(n,k) = binomial(n+k,2*k) + 2*binomial(n+k,2*k+1).
The row generating polynomials P(n,x) are a generalization of the Morgan-Voyce polynomials b(n,x) and B(n,x). They satisfy the recurrence equation P(n,x) = (x+2)*P(n-1,x) - P(n-2,x) for n >= 2, with initial conditions P(0,x) = 1, P(1,x) = x+r+1 and with r = 2. The cases r = 0 and r = 1 give the Morgan-Voyce polynomials A085478 and A078812 respectively. Andre-Jeannin has considered the case of general r.
P(n,x) = U(n+1,1+x/2) + U(n,1+x/2), where U(n,x) denotes the Chebyshev polynomial of the second kind - see A053117. P(n,x) = (2/x)*(T(2*n+2,u)-T(2*n,u)), where u = sqrt((x+4)/4) and T(n,x) denotes the Chebyshev polynomial of the first kind - see A053120. P(n,x) = Product_{k = 1..n} ( x + 4*(sin(k*Pi/(2*n+1)))^2 ). P(n,x) = 1/x*(b(n+1,x) - b(n-1,x)) and P(n,x) = 1/x*{(b(2*n+2,x)+1)/b(n+1,x) - (b(2*n,x)+1)/b(n,x)}, where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478. Cf. A211957.
(End)
From Wolfdieter Lang, Oct 18 2012 (Start)
O.g.f. column No. s: ((1+x)/(1-x)^2)*(x/(1-x)^2)^s, s >= 0. (from the Riordan data given in a comment above).
O.g.f. of the row polynomials R(k,x):= Sum_{s=0..k} ( T(k,s)*x^s ), k>=0: (1+z)/(1-(2+x)*z+z^2) (from the Riordan property).
(End)
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 12 2013

More terms from Paul Barry, Oct 17 2005

A053122 Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).

Original entry on oeis.org

1, -2, 1, 3, -4, 1, -4, 10, -6, 1, 5, -20, 21, -8, 1, -6, 35, -56, 36, -10, 1, 7, -56, 126, -120, 55, -12, 1, -8, 84, -252, 330, -220, 78, -14, 1, 9, -120, 462, -792, 715, -364, 105, -16, 1, -10, 165, -792, 1716, -2002, 1365, -560, 136, -18, 1, 11, -220, 1287, -3432, 5005, -4368, 2380, -816, 171, -20

Offset: 0

Wolfdieter Lang

Apart from signs, identical to A078812.
Another version with row-leading 0's and differing signs is given by A285072.
G.f. for row polynomials S(n,x-2) (signed triangle): 1/(1+(2-x)*z+z^2). Unsigned triangle |a(n,m)| has g.f. 1/(1-(2+x)*z+z^2) for row polynomials.
Row sums (signed triangle) A049347(n) (periodic(1,-1,0)). Row sums (unsigned triangle) A001906(n+1)=F(2*(n+1)) (even-indexed Fibonacci).
In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The (unsigned) column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for m=0..5, resp. For m=6..23 they are A010966..(+2)..A011000 and for m=24..49 they are A017713..(+2)..A017763.
Riordan array (1/(1+x)^2,x/(1+x)^2). Inverse array is A039598. Diagonal sums have g.f. 1/(1+x^2). - Paul Barry, Mar 17 2005. Corrected by Wolfdieter Lang, Nov 13 2012.
Unsigned version is in A078812. - Philippe Deléham, Nov 05 2006
Also row n gives (except for an overall sign) coefficients of characteristic polynomial of the Cartan matrix for the root system A_n. - Roger L. Bagula, May 23 2007
From Wolfdieter Lang, Nov 13 2012: (Start)
The A-sequence for this Riordan triangle is A115141, and the Z-sequence is A115141(n+1), n>=0. For A- and Z-sequences for Riordan matrices see the W. Lang link under A006232 with details and references.
S(n,x^2-2) = sum(r(j,x^2),j=0..n) with Chebyshev's S-polynomials and r(j,x^2) := R(2*j+1,x)/x, where R(n,x) are the monic integer Chebyshv T-polynomials with coefficients given in A127672. Proof from comparing the o.g.f. of the partial sum of the r(j,x^2) polynomials (see a comment on the signed Riordan triangle A111125) with the present Riordan type o.g.f. for the row polynomials with x -> x^2.  (End)
S(n,x^2-2) = S(2*n+1,x)/x, n >= 0, from the odd part of the bisection of the o.g.f. - Wolfdieter Lang, Dec 17 2012
For a relation to a generator for the Narayana numbers A001263, see A119900, whose columns are unsigned shifted rows (or antidiagonals) of this array, referring to the tables in the example sections. - Tom Copeland, Oct 29 2014
The unsigned rows of this array are alternating rows of a mirrored A011973 and alternating shifted rows of A030528 for the Fibonacci polynomials. - Tom Copeland, Nov 04 2014
Boas-Buck type recurrence for column k >= 0 (see Aug 10 2017 comment in A046521 with references): a(n, m) =  (2*(m + 1)/(n - m))*Sum_{k = m..n-1} (-1)^(n-k)*a(k, m), with input a(n, n) = 1, and a(n,k) = 0 for n < k. - Wolfdieter Lang, Jun 03 2020
Row n gives the characteristic polynomial of the (n X n)-matrix M where M[i,j] = 2 if i = j, -1 if |i-j| = 1 and 0 otherwise. The matrix M is positive definite and has 2-condition number (cot(Pi/(2*n+2)))^2. - Jianing Song, Jun 21 2022
Also the convolution triangle of (-1)^(n+1)*n. - Peter Luschny, Oct 07 2022

			The triangle a(n,m) begins:
n\m   0    1    2     3     4     5     6    7    8  9
0:    1
1:   -2    1
2:    3   -4    1
3:   -4   10   -6     1
4:    5  -20   21    -8     1
5:   -6   35  -56    36   -10     1
6:    7  -56  126  -120    55   -12     1
7:   -8   84 -252   330  -220    78   -14    1
8:    9 -120  462  -792   715  -364   105  -16    1
9:  -10  165 -792  1716 -2002  1365  -560  136  -18  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 13 2012
E.g., fourth row (n=3) {-4,10,-6,1} corresponds to the polynomial S(3,x-2) = -4+10*x-6*x^2+x^3.
From _Wolfdieter Lang_, Nov 13 2012: (Start)
Recurrence: a(5,1) = 35 = 1*5 + (-2)*(-20) -1*(10).
Recurrence from Z-sequence [-2,-1,-2,-5,...]: a(5,0) = -6 = (-2)*5 + (-1)*(-20) + (-2)*21 + (-5)*(-8) + (-14)*1.
Recurrence from A-sequence [1,-2,-1,-2,-5,...]: a(5,1) = 35 = 1*5  + (-2)*(-20) + (-1)*21 + (-2)*(-8) + (-5)*1.
(End)
E.g., the fourth row (n=3) {-4,10,-6,1} corresponds also to the polynomial S(7,x)/x = -4 + 10*x^2 - 6*x^4 + x^6. - _Wolfdieter Lang_, Dec 17 2012
Boas-Buck type recurrence: -56 = a(5, 2) = 2*(-1*1 + 1*(-6) - 1*21) = -2*28 = -56. - _Wolfdieter Lang_, Jun 03 2020

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62.
Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S.: ISBN 0-8218-2848-7, 1978, p. 463.

T. D. Noe, Rows n=0..50 of triangle, flattened
Wolfdieter Lang, First rows of the triangle.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
P. Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Naiomi T. Cameron and Asamoah Nkwanta, On some (pseudo) involutions in the Riordan group, J. of Integer Sequences, 8(2005), 1-16.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014 (p. 10). - _Tom Copeland_, Oct 11 2014
Pentti Haukkanen, Jorma Merikoski, Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
Eric Weisstein's World of Mathematics, Cartan Matrix
Eric Weisstein's World of Mathematics, Dynkin Diagram
Index entries for sequences related to Chebyshev polynomials.

Cf. A005248, A127677, A053123, A049310.
Cf. A011973, A030528, A034867, A119900.
Cf. A285072 (version with row-leading 0's and differing signs). - Eric W. Weisstein, Apr 09 2017

seq(seq((-1)^(n+m)*binomial(n+m+1,2*m+1),m=0..n),n=0..10); # Robert Israel, Oct 15 2014
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> -(-1)^n*n); # Peter Luschny, Oct 07 2022

T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] (* Roger L. Bagula, May 23 2007 *)
(* Alternative code for the matrices from MathWorld: *)
sln[n_] := 2IdentityMatrix[n] - PadLeft[PadRight[IdentityMatrix[n - 1], {n, n - 1}], {n, n}] - PadLeft[PadRight[IdentityMatrix[n - 1], {n - 1, n}], {n, n}] (* Roger L. Bagula, May 23 2007 *)

@CachedFunction
def A053122(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A053122(n-1,k-1)-A053122(n-2,k)-2*A053122(n-1,k)
for n in (0..9): [A053122(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

a(n, m) := 0 if n

a(n, m) = -2*a(n-1, m) + a(n-1, m-1) - a(n-2, m), a(n, -1) := 0 =: a(-1, m), a(0, 0)=1, a(n, m) := 0 if n

O.g.f. for m-th column (signed triangle): ((x/(1+x)^2)^m)/(1+x)^2.

From Jianing Song, Jun 21 2022: (Start)

T(n,k) = [x^k]f_n(x), where f_{-1}(x) = 0, f_0(x) = 1, f_n(x) = (x-2)*f_{n-1}(x) - f_{n-2}(x) for n >= 2.

f_n(x) = (((x-2+sqrt(x^2-4*x))/2)^(n+1) - ((x-2-sqrt(x^2-4*x))/2)^(n+1))/sqrt(x^2-4x).

The roots of f_n(x) are 2 + 2*cos(k*Pi/(n+1)) = 4*(cos(k*Pi/(2*n+2)))^2 for 1 <= k <= n. (End)

A263916 Coefficients of the Faber partition polynomials.

Original entry on oeis.org

-1, -2, 1, -3, 3, -1, -4, 4, 2, -4, 1, -5, 5, 5, -5, -5, 5, -1, -6, 6, 6, -6, 3, -12, 6, -2, 9, -6, 1, -7, 7, 7, -7, 7, -14, 7, -7, -7, 21, -7, 7, -14, 7, -1, -8, 8, 8, -8, 8, -16, 8, 4, -16, -8, 24, -8, -8, 12, 24, -32, 8, 2, -16, 20, -8, 1

Offset: 1

Tom Copeland, Oct 29 2015

The coefficients of the Faber polynomials F(n,b(1),b(2),...,b(n)) (Bouali, p. 52) in the order of the partitions of Abramowitz and Stegun. Compare with A115131 and A210258.
These polynomials occur in discussions of the Virasoro algebra, univalent function spaces and the Schwarzian derivative, symmetric functions, and free probability theory. They are intimately related to symmetric functions, free probability, and Appell sequences through the raising operator R = x - d log(H(D))/dD for the Appell sequence inverse pair associated to the e.g.f.s H(t)e^(xt) (cf. A094587) and (1/H(t))e^(xt) with H(0)=1.
Instances of the Faber polynomials occur in discussions of modular invariants and modular functions in the papers by Asai, Kaneko, and Ninomiya, by Ono and Rolen, and by Zagier. - Tom Copeland, Aug 13 2019
The Faber polynomials, denoted by s_n(a(t)) where a(t) is a formal power series defined by a product formula, are implicitly defined by equation 13.4 on p. 62 of Hazewinkel so as to extract the power sums of the reciprocals of the zeros of a(t). This is the Newton identity expressing the power sum symmetric polynomials in terms of the elementary symmetric polynomials/functions. - Tom Copeland, Jun 06 2020
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)

			F(1,b1) = - b1
F(2,b1,b2) = -2 b2 + b1^2
F(3,b1,b2,b3) = -3 b3 + 3 b1 b2 - b1^3
F(4,b1,...) = -4 b4 + 4 b1 b3 + 2 b2^2  - 4 b1^2 b2 + b1^4
F(5,...) = -5 b5 + 5 b1 b4 + 5 b2 b3 - 5 b1^2 b3 - 5 b1 b2^2 + 5 b1^3 b2 - b1^5
------------------------------
IF(1,b1) = -b1
IF(2,b1,,b2) = -b2 + b1^2
IF(3,b1,b2,b3) = -2 b3 + 3 b1 b2 - b1^3
IF(4,b1,...) = -6 b4 + 8 b1 b3 + 3 b2^2  - 6 b1^2 b2 + b1^4
IF(5,...) = -24 b5 + 30 b1 b4 + 20 b2 b3 - 20 b1^2 b3 - 15 b1 b2^2 + 10 b1^3 b2 - b1^5
------------------------------
For 1/(1+x)^2 = 1- 2x + 3x^2 - 4x^3 + 5x^4 - ..., F(n,-2,3,-4,...) = (-1)^(n+1) 2.
------------------------------
F(n,x,2x,...,nx), F(n,-x,2x,-3x,...,(-1)^n n*x), and F(n,(2-x),1,0,0,...) are related to the Chebyshev polynomials through A127677 and A111125. See also A110162, A156308, A208513, A217476, and A220668.
------------------------------
For b1 = p, b2 = q, and all other indeterminates 0, see A113279 and A034807.
For b1 = -y, b2 = 1 and all other indeterminates 0, see A127672.

H. Airault, "Symmetric sums associated to the factorization of Grunsky coefficients," in Groups and Symmetries: From Neolithic Scots to John McKay, CRM Proceedings and Lecture Notes: Vol. 47, edited by J. Harnad and P. Winternitz, American Mathematical Society, 2009.
D. Bleeker and B. Booss, Index Theory with Applications to Mathematics and Physics, International Press, 2013, (see section 16.7 Characteristic Classes and Curvature).
M. Hazewinkel, Formal Groups and Applications, Academic Press, New York San Francisco London, 1978, p. 120.
F. Hirzebruch, Topological methods in algebraic geometry. Second, corrected printing of the third edition. Die Grundlehren der Mathematischen Wissenschaften, Band 131 Springer-Verlag, Berlin Heidelberg New York, 1978, p. 11 and 92.
D. Knutson, λ-Rings and the Representation Theory of the Symmetric Group, Lect. Notes in Math. 308, Springer-Verlag, 1973, p. 35.
D. Yau, Lambda-Rings, World Scientific Publishing Co., Singapore, 2010, p. 45.

Vincenzo Librandi, Table of n, a(n) for n = 1..5762
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
H. Airault, Remarks on Faber polynomials, International Mathematical Forum, 3, no. 9, 2008, pages 449-456.
H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, pages 179-222.
H. Airault and Y. Neretin, On the action of the Virasoro algebra on the space of univalent functions, arXiv:0704.2149 [math.RT], 2007.
F. Ardila, Algebraic and geometric methods in enumerative combinatorics, arXiv:1409.2562 [math.CO], 2015.
T. Asai, M. Kaneko, and H. Ninomiya, Zeros of certain modular functions and an application, 1997.
A. Bouali, Faber polynomials Cayley-Hamilton equation and Newton symmetric functions, Bulletin des Sciences Mathématiques, Volume 130, Issue 1, Jan-Feb 2006, pages 49-70.
J. M. Campbell, Combinatorial interpretations of primitivity in the algebra of symmetric functions, Sarajevo J. Math., 17(30) (2021), 151--165.
P. Cartier, Mathemagics: A tribute to L. Euler and R. Feynman, Séminaire Lotharingien de Combinatoire 44 (2000), Article B44d, 2000, p. 53.
P. Cartier, A primer of Hopf algebras, preprint, Institut des Hautes Etudes Scientifiques, France, 2006, pp. 56 and 57.
V. Chan, Topological K-theory of complex projective spaces, senior's thesis UC Davis (p. 11 on Chern characters of complex vector bundles), 2013.
Gi-Sang Cheon, Hana Kim, and Louis W. Shapiro, An algebraic structure for Faber polynomials, Lin. Alg. Applic. 433 (2010) 1170-1179.
Tom Copeland, Lagrange a la Lah, 2011.
Tom Copeland, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants, MathOverflow question, 2012.
Tom Copeland, Addendum to Elliptic Lie Triad, 2015.
Tom Copeland, Connection between the Chebyshev polynomials and the Faber polynomials, MathOverflow question, 2015.
Tom Copeland, The Faber Appells, 2020.
Tom Copeland, Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials, 2020.
A. Dress and C. Siebeneicher, The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, lambda-rings, and the universal ring of Witt vectors, Adv. in Math., Vol. 78, Issue 1, Nov. 1989, pages 1-41.
D. Dugger, A Geometric Introduction to K-Theory, p. 288 (called Newton polynomials).
M. Eiermann and R. Varga, Zeros and local extreme points of Faber polynomials associated with hypocycloidal domains, Elect. Trans. on Numer. Analysis, Vol. 1, p. 49-71, 1993.
H. Figueroa and J. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (normalized versions on pp. 42 and 78, denoted as Schur polynomials).
R. Friedrich and J. McKay, Formal groups, Witt vectors and free probability, arXiv:1204.6522 [math.OA], 2012.
A. Hatcher, Vector Bundles and K-Theory, Version 2.2, 2017, p. 63.
M. Hazewinkel, Three lectures on formal group laws, Canadian Mathematical Society Conference Proceedings, Vol. 5, p. 51-67, 1986.
Y. He, The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning, arXiv:1812.02893 [hep-th], 2018, p. 146.
O. Iena, On symbolic computations with Chern classes: Remarks on the library CHERN.LIB for SINGULAR, p. 6.
B. Konopelchenko, Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022 [nlin.SI], 2008.
R. Lu, Regularized equivariant Euler classes and gamma functions, Ph.D. thesis, Dept. of Pure Mathematics, University of Adelaide, 2008, p. 86.
K. Maslanka, Effective method of computing Li’S coefficients and their propertie, arXiv:0402168v5 [math.NT], 2004, p. 6.
MathOverflow, Canonical reference for Chern characteristic classes, a question posed by Tom Copeland, 2019.
J. McKay and A. Sebbar, On replicable functions: an introduction, Frontiers in Number Theory, Physics, and Geometry II, pp. 373-386.
L. Nicolaescu, Lectures on the Geometry of Manifolds, p.337, 2018.
J. Novak and M. LaCroix, Three lectures on free probability, arXiv:1205.2097 [math.CO], 2012.
K. Ono and L. Rolen, On Witten's extremal partition functions , arXiv:1807.00444 [math.NT], 2019.
T. Takebe, Lee-Peng Teo, and A. Zabrodin, Löwner equations and dispersionless hierarchies, arXiv:math/0605161 [math.CV], p. 24, 2006.
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Wikipedia, Newton identities.
D. Zagier, Traces of singular moduli, 2011.

Cf. Row sums of absolute values: A000225.
Cf. A034807, A036039, A036040, A094587, A110162, A111125, A113279, A115131.
Cf. A127671, A127672, A127677, A130561, A133314, A156308, A208513, A210258.
Cf. A217476, A220668.
Cf. A263634.

F[0] = 1; F[1] = -b[1]; F[2] = b[1]^2 - 2 b[2]; F[n_] := F[n] = -b[1] F[n - 1] - Sum[b[n - k] F[k], {k, 1, n - 2}] - n b[n] // Expand;
row[n_] := (List @@ F[n]) /. b[_] -> 1 // Reverse;
Table[row[n], {n, 1, 8}] // Flatten // Rest (* Jean-François Alcover, Jun 12 2017 *)

-log(1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n.
-d(1 + b(1) x + b(2) x^2 + ...)/dx / (1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) x^(n-1).
F(n,b(1),...,b(n)) = -n*b(n) - Sum_{k=1..n-1} b(n-k)*F(k,b(1),...,b(k)).
Umbrally, with B(x) = 1 + b(1) x + b(2) x^2 + ..., B(x) = exp[log(1-F.x)] and 1/B(x) = exp[-log(1-F.x)], establishing a connection to the e.g.f. of A036039 and the symmetric polynomials.
The Stirling partition polynomials of the first kind St1(n,b1,b2,...,bn;-1) = IF(n,b1,b2,...,bn) (cf. the Copeland link Lagrange a la Lah, signed A036039, and p. 184 of Airault and Bouali), i.e., the cyclic partition polynomials for the symmetric groups, and the Faber polynomials form an inverse pair for isolating the indeterminates in their definition, for example, F(3,IF(1,b1),IF(2,b1,b2)/2!,IF(3,b1,b2,b3)/3!)= b3, with bk = b(k), and IF(3,F(1,b1),F(2,b1,b2),F(3,b1,b2,b3))/3!= b3.
The polynomials specialize to F(n,t,t,...) = (1-t)^n - 1.
See Newton Identities on Wikipedia on relation between the power sum symmetric polynomials and the complete homogeneous and elementary symmetric polynomials for an expression in multinomials for the coefficients of the Faber polynomials.
(n-1)! F(n,x[1],x[2]/2!,...,x[n]/n!) = - p_n(x[1],...,x[n]), where p_n are the cumulants of A127671 expressed in terms of the moments x[n]. - Tom Copeland, Nov 17 2015
-(n-1)! F(n,B(1,x[1]),B(2,x[1],x[2])/2!,...,B(n,x[1],...,x[n])/n!) = x[n] provides an extraction of the indeterminates of the complete Bell partition polynomials B(n,x[1],...,x[n]) of A036040. Conversely, IF(n,-x[1],-x[2],-x[3]/2!,...,-x[n]/(n-1)!) = B(n,x[1],...,x[n]). - Tom Copeland, Nov 29 2015
For a square matrix M, determinant(I - x M) = exp[-Sum_{k>0} (trace(M^k) x^k / k)] = Sum_{n>0} [ P_n(-trace(M),-trace(M^2),...,-trace(M^n)) x^n/n! ] = 1 + Sum_{n>0} (d[n] x^n), where P_n(x[1],...,x[n]) are the cycle index partition polynomials of A036039 and d[n] = P_n(-trace(M),-trace(M^2),...,-trace(M^n)) / n!. Umbrally, det(I - x M)= exp[log(1 - b. x)] = exp[P.(-b_1,..,-b_n)x] = 1 / (1-d.x), where b_k = tr(M^k). Then F(n,d[1],...,d[n]) = tr[M^n]. - Tom Copeland, Dec 04 2015
Given f(x) = -log(g(x)) = -log(1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n, action on u_n = F(n,b(1),...,b(n)) with A133932 gives the compositional inverse finv(x) of f(x), with F(1,b(1)) not equal to zero, and f(g(finv(x))) = f(e^(-x)). Note also that exp(f(x)) = 1 / g(x) = exp[Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n] implies relations among A036040, A133314, A036039, and the Faber polynomials. - Tom Copeland, Dec 16 2015
The Dress and Siebeneicher paper gives combinatorial interpretations and various relations that the Faber polynomials must satisfy for integral values of its arguments. E.g., Eqn. (1.2) p. 2 implies [2 * F(1,-1) + F(2,-1,b2) + F(4,-1,b2,b3,b4)]  mod(4) = 0. This equation implies that [F(n,b1,b2,...,bn)-(-b1)^n] mod(n) = 0 for n prime. - Tom Copeland, Feb 01 2016
With the elementary Schur polynomials S(n,a_1,a_2,...,a_n) = Lah(n,a_1,a_2,...,a_n) / n!, where Lah(n,...) are the refined Lah polynomials of A130561, F(n,S(1,a_1),S(2,a_1,a_2),...,S(n,a_1,...,a_n)) = -n * a_n since sum_{n > 0} a_n x^n = log[sum{n >= 0} S(n,a_1,...,a_n) x^n]. Conversely, S(n,-F(1,a_1),-F(2,a_1,a_2)/2,...,-F(n,a_1,...,a_n)/n) = a_n. - Tom Copeland, Sep 07 2016
See Corollary 3.1.3 on p. 38 of Ardila and Copeland's two MathOverflow links to relate the Faber polynomials, with arguments being the signed elementary symmetric polynomials, to the logarithm of determinants, traces of powers of an adjacency matrix, and number of walks on graphs. - Tom Copeland, Jan 02 2017
The umbral inverse polynomials IF appear on p. 19 of Konopelchenko as partial differential operators. - Tom Copeland, Nov 19 2018

More terms from Jean-François Alcover, Jun 12 2017

A127677 Scaled coefficient table for Chebyshev polynomials 2T(2n, sqrt(x)/2) (increasing even scaled powers, without zero entries).

Original entry on oeis.org

2, -2, 1, 2, -4, 1, -2, 9, -6, 1, 2, -16, 20, -8, 1, -2, 25, -50, 35, -10, 1, 2, -36, 105, -112, 54, -12, 1, -2, 49, -196, 294, -210, 77, -14, 1, 2, -64, 336, -672, 660, -352, 104, -16, 1, -2, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 2, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 0

Wolfdieter Lang, Mar 07 2007

2*T(2*n,x) = Sum_{m=0..n} a(n,m)*(2*x)^(2*m).
Closely related to A284982, which has opposite signs and rows begin with 0 of alternating signs instead of +/2. - Eric W. Weisstein, Apr 07 2017
Bisection triangle of A127672 (without zero entries, even part). The odd part is ((-1)^(n-m))*A111125(n,m).
If the leading 2 is replaced by a 1 we get the essentially identical sequence A110162. - N. J. A. Sloane, Jun 09 2007
Also row n gives coefficients of characteristic polynomial of the Cartan matrix for the root system B_n (or, equally, C_n). - Roger L. Bagula, May 23 2007
From Wolfdieter Lang, Oct 04 2013: (Start)
This triangle a(n,m) is used to express the length ratio side/R given by s(4*n+2) = 2*sin(Pi/(4*n+2)) = 2*cos(2*n*Pi/(4*n+2)) in a regular (4*n+2)-gon, inscribed in a circle with radius R, in terms of rho(4*n+2) = 2*cos(Pi/4*n+2), the length ratio of (the smallest diagonal)/side (for n=2 there is no such diagonal).
s(4*n+2) = Sum_{m=0..n}a(n,m)*rho(4*n+2)^(2*m). This formula is needed to show that the total sum of all length ratios in a (4*n+2)-gon is an integer in the algebraic number field Q(rho(4*n+2)). Note that rho(4*n+2) has degree delta(4*n+2) = A055034(4*n+2). Therefore one has to take s(4*n+2) modulo C(4*n+2, x=rho(4*n+2)), the minimal polynomial of rho(4*n+2) (see A187360). Thanks go to Seppo Mustonen for asking me to look into this problem. See ((-1)^(n-m))*A111125(n,m) for the (4*n)-gon situation. (End)

			The triangle a(n,m) starts:
n\m  0    1    2     3     4     5     6     7    8   9  10 ...
0:   2
1:  -2    1
2:   2   -4    1
3:  -2    9   -6     1
4:   2  -16   20    -8     1
5:  -2   25  -50    35   -10     1
6:   2  -36  105  -112    54   -12     1
7:  -2   49 -196   294  -210    77   -14     1
8:   2  -64  336  -672   660  -352   104   -16    1
9:  -2   81 -540  1386 -1782  1287  -546   135  -18   1
10:  2 -100  825 -2640  4290 -4004  2275  -800  170 -20  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 21 2012.
n=3: [-2,9,-6,1] stands for -2*1 + 9*(2*x)^2 -6*(2*x)^4 +1*(2*x)^6 = 2*(1+18*x^2-48*x^4+32*x^6) = 2*T(6,x).
(4*n+2)-gon side/radius s(4*n+2) as polynomial in rho(4*n+2) = smallest diagonal/side: n=0: s(2) = 2 (rho(2)=0); n=1: s(6) = -2 + rho(6)^2 = -2 + 3 = 1, (C(6,x) = x^2 - 3); n=2: s(10) = 2 - 4*rho(10)^2 + 1*rho(10)^4 = 2 - 4*rho(10)^2 + (5*rho(10)^2 - 5) = -3 + rho(10)^2, (C(10,x) = x^4 - 5*x^2 + 5). - _Wolfdieter Lang_, Oct 04 2013

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62
Sigurdur Helgasson,Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978,p. 463.

Wolfdieter Lang, First 10 rows and more.
P. Damianou , On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
P. Damianou and C. Evripidou, Characteristic and Coxeter polynomials for affine Lie algebras, arXiv preprint arXiv:1409.3956 [math.RT], 2014.
Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart., Nov. 2005, pp. 359-370. See Table 1.6 (an unsigned version).
Eric Weisstein's World of Mathematics, Cartan Matrix
Eric Weisstein's World of Mathematics, Dynkin Diagram

Cf. A284982 (opposite signs and rows begin with 0).
Row sums (signed): -A061347(n+3) for n>=0.
Row sums (unsigned): A005248(n) = L(2*n), where L=Lucas.
Cf. A005248, A053122.
Cf. A263916, A111125, A208513, A034807, A127672.

T[n_, m_, d_] := If[ n == m, 2, If[n == d && m == d - 1, -2, If[(n == m - 1 || n == m + 1), -1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] a = Join[M[1], Table[CoefficientList[CharacteristicPolynomial[M[d], x], x], {d, 1, 10} ]] (* Roger L. Bagula, May 23 2007 *)
CoefficientList[2 ChebyshevT[2 Range[0, 10], Sqrt[x]/2], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[Table[(-1)^n LucasL[2 n, Sqrt[-x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)

a(n,m) = {if(n>=2, -2*a(n-1,m)+a(n-1,m-1)-a(n-2,m), if(n==0, if(m!=0,0,2), if(m==0,-2, if(m==1,1,0))))};
for(n=0,10,for(m=0,n,print1(a(n,m),", "))) \\ Hugo Pfoertner, Jul 19 2020

a(n,m) = 0 if n < m; a(n,0) = 2*(-1)^n; a(n,m) = ((-1)^(n+m))*n*binomial(n+m-1, 2*m-1)/m.
a(n,m) = 0 if n < m, a(0,0) = 2, a(n,m) = (-1)^(n-m)*(2*n/(n+m))*binomial(n+m, n-m), n >= 1. From Waring's formula applied to Chebyshev's T-polynomials. See also A110162. - Wolfdieter Lang, Nov 21 2012
The o.g.f. Sum_{n>=0} p(n,x)*z^n, n>=0, for the row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m is (2 + z*(2-x))/((z+1)^2 - z*x). Here p(n,x) = R(2*n,sqrt(x)) := 2*T(2*n,sqrt(x)/2) with Chebyshev's T-polynomials. For the R-polynomials see A127672. - Wolfdieter Lang, Nov 28 2012
From Tom Copeland, Nov 07 2015: (Start)
A logarithmic generator is 2*(1-log(1+x))-log(1-t*x/(1+x)^2) = 2 - log(1+(2-t)*x+x^2) = 2 + (-2 + t)*x + (2 - 4*t + t^2) x^2/2 + (-2 + 9*t - 6*t^2 + t^3) x^3/3 + ..., so a number of relations to the Faber polynomials of A263916 hold with p(0,x) = 2:
1) p(n,x) = F(n,(2-x),1,0,0,..)
2) p(n,x) = (-1)^n 2 + F(n,-x,2x,-3x,...,(-1)^n n*x)
3) p(n,x) = (-1)^n [2 + F(n,x,2x,3x,...,n*x)].
The unsigned array contains the partial sums of A111125 modified by appending a column of zeros, except for an initial two, to A111125. Then the difference of consecutive rows of unsigned A127677, further modified by appending an initial rows of zeros, generates the modified A111125. Cf. A208513 and A034807.
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).
See A111125 for a relation to the squares of the odd row polynomials here with the constant removed.
p(n,x)^2 = 2 + p(2*n,x). See also A127672. (End)
a(n,m) = -2*a(n-1,m) + a(n-1,m-1) - a(n-2,m) for n >= 2 with initial conditions a(0,0) = 2, a(1,0) = -2, a(1,1) = 1, a(0,m) = 0 for m != 0, a(1,m) = 0 for m != 0,1. - William P. Orrick, Jun 09 2020
p(n,x) = (x-2)*p(n-1,x) - p(n-2,x) for n >= 2. - William P. Orrick, Jun 09 2020

Definition corrected by Eric W. Weisstein, Apr 06 2017

A179260 Decimal expansion of the connective constant of the honeycomb lattice.

Original entry on oeis.org

1, 8, 4, 7, 7, 5, 9, 0, 6, 5, 0, 2, 2, 5, 7, 3, 5, 1, 2, 2, 5, 6, 3, 6, 6, 3, 7, 8, 7, 9, 3, 5, 7, 6, 5, 7, 3, 6, 4, 4, 8, 3, 3, 2, 5, 1, 7, 2, 7, 2, 8, 4, 9, 7, 2, 2, 3, 0, 1, 9, 5, 4, 6, 2, 5, 6, 1, 0, 7, 0, 0, 1, 5, 0, 0, 2, 2, 0, 4, 7, 1, 7, 4, 2, 9, 6, 7, 9, 8, 6, 9, 7, 0, 0, 6, 8, 9, 1, 9, 2

Offset: 1

Jonathan Vos Post, Jul 06 2010

This is the case n=8 of the ratio Gamma(1/n)*Gamma((n-1)/n)/(Gamma(2/n)*Gamma((n-2)/n)). - Bruno Berselli, Dec 13 2012
An algebraic integer of degree 4: largest root of x^4 - 4x^2 + 2. - Charles R Greathouse IV, Nov 05 2014
This number is also the length ratio of the shortest diagonal (not counting the side) of the octagon and the side. This ratio is A121601 for the longest diagonal. - Wolfdieter Lang, May 11 2017 [corrected Oct 28 2020]
From Wolfdieter Lang, Apr 29 2018: (Start)
This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 3, pp. 69-74. See also the comments in A302711 with the Romanus link and his Exemplum tertium.
This problem is equivalent to R(45, 2*sin(Pi/120)) = 2*sin(3*Pi/8) with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/120) see A302715. (End)

			1.84775906502257351225636637879357657364483325172728497223019546256107001500...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.10, p. 333.
Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.
Neal Madras and Gordon Slade, Self-avoiding walks, Probability and its Applications, Birkhäuser Boston, Inc. Boston, MA, 1993.

Hugo Duminil-Copin and Stanislav Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt2), arXiv:1007.0575 [math-ph], 2011.
Hugo Duminil-Copin and Stanislav Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt2), Ann. Math. 175 (2012), pp. 1653-1665.
Steven Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 5.10; arXiv:2001.00578 [math.HO], 2020.
Pierre-Louis Giscard, Que sait-on compter sur un graphe. Partie 3 (in French), Images des Mathématiques, CNRS, 2020.
Gregory F. Lawler, Oded Schramm, and Wendelin Werner, On the scaling limit of planar self-avoiding walk, Fractal Geometry and applications: a jubilee of Benoit Mandelbrot, Part 2, 339-364. Proc.
Bernard Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Phys. Rev. Lett. 49 (1982), 1062-1065.
Jonathan Sondow and Huang Yi, New Wallis- and Catalan-type infinite products for Pi, e, and sqrt(2+sqrt(2)), arXiv:1005.2712 [math.NT], 2010.
Jonathan Sondow and Huang Yi, New Wallis- and Catalan-type infinite products for Pi, e, and sqrt(2+sqrt(2)), Amer. Math. Monthly 117 (2010) 912-917.
Index entries for algebraic numbers, degree 4.
Index entries for sequences related to Chebyshev polynomials.

Cf. A002193, A047522, A101464, A127672, A144981, A154739, A302715, A121601.

RealDigits[Sqrt[2+Sqrt[2]],10,120][[1]] (* Harvey P. Dale, Jan 19 2014 *)

sqrt(2+sqrt(2)) \\ Charles R Greathouse IV, Nov 05 2014

sqrt(2+sqrt(2)) = (2/1)(6/7)(10/9)(14/15)(18/17)(22/23)... (see Sondow-Yi 2010).
Equals 1/A154739. - R. J. Mathar, Jul 11 2010
Equals 2*A144981. - Paul Muljadi, Aug 23 2010
log (A001668(n)) ~ n log k where k = sqrt(2+sqrt(2)). - Charles R Greathouse IV, Nov 08 2013
2*cos(Pi/8) = sqrt(2+sqrt(2)). See a remark on the smallest diagonal in the octagon above. - Wolfdieter Lang, May 11 2017
Equals also 2*sin(3*Pi/8). See the comment on van Roomen's third problem above. - Wolfdieter Lang, Apr 29 2018
Equals i^(1/4) + i^(-1/4). - Gary W. Adamson, Jul 06 2022
Equals Product_{k>=0} ((8*k + 2)*(8*k + 6))/((8*k + 1)*(8*k + 7)). - Antonio Graciá Llorente, Feb 24 2024
Equals Product_{k>=1} (1 - (-1)^k/A047522(k)). - Amiram Eldar, Nov 22 2024

A193678 Discriminant of Chebyshev C-polynomials.

Original entry on oeis.org

1, 8, 108, 2048, 50000, 1492992, 52706752, 2147483648, 99179645184, 5120000000000, 292159150705664, 18260173718028288, 1240576436601868288, 91029559914971267072, 7174453500000000000000, 604462909807314587353088, 54214017802982966177103872

Offset: 1

Wolfdieter Lang, Aug 07 2011

The array of coefficients of the (monic) Chebyshev C-polynomials is found under A127672 (where they are called, in analogy to the S-polynomials, R-polynomials).
See A127670 for the formula in terms of the square of a Vandermonde determinant, where now the zeros are xn[j]:=2*cos(Pi*(2*j+1)/(2*n)), j=0,..,n-1.
One could add a(0)=0 for the discriminant of C(0,x)=2.
Except for sign, a(n) is the field discriminant of 2^(1/n); see the Mathematica program. - Clark Kimberling, Aug 03 2015

			n=3: The zeros are [sqrt(3),0,-sqrt(3)]. The Vn(xn[0],..,xn[n-1]) matrix is [[1,1,1],[sqrt(3),0,-sqrt(3)],[3,0,3]]. The squared determinant is 108 = a(3).

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.

Robert Israel, Table of n, a(n) for n = 1..320
Sinan Deveci, On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function, arXiv:2211.10751 [math.NT], 2022.

Cf. A127670.

[(2^(n-1))*n^n: n in [1..20]]; // Vincenzo Librandi, Aug 04 2015

seq(discrim(2*orthopoly[T](n,x/2), x), n = 1..50); # Robert Israel, Aug 04 2015

t=Table[NumberFieldDiscriminant[2^(1/m)], {m, 1, 20}] (* signed version *)
Abs[t] (* Clark Kimberling, Aug 03 2015 *)
Table[(2^(n - 1)) n^n, {n, 20}] (* Vincenzo Librandi, Aug 04 2015 *)

a(n) = (Det(Vn(xn[0],..,xn[n-1])))^2, with the n x n Vandermonde matrix Vn and the zeros xn[j],j=0..n-1, given above in a comment.
a(n) = (2^(n-1))*n^n, n>=1.
a(n) = A000079(n-1)*A000312(n). - Omar E. Pol, Aug 27 2011

A130777 Coefficients of first difference of Chebyshev S polynomials.

Original entry on oeis.org

1, -1, 1, -1, -1, 1, 1, -2, -1, 1, 1, 2, -3, -1, 1, -1, 3, 3, -4, -1, 1, -1, -3, 6, 4, -5, -1, 1, 1, -4, -6, 10, 5, -6, -1, 1, 1, 4, -10, -10, 15, 6, -7, -1, 1, -1, 5, 10, -20, -15, 21, 7, -8, -1, 1, -1, -5, 15, 20, -35, -21, 28, 8, -9, -1, 1, 1, -6, -15, 35, 35, -56, -28, 36, 9, -10, -1, 1

Offset: 0

Philippe Deléham, Jul 14 2007

Inverse of triangle in A061554.
Signed version of A046854.
From Paul Barry, May 21 2009: (Start)
Riordan array ((1-x)/(1+x^2),x/(1+x^2)).
This triangle is the coefficient triangle for the Hankel transforms of the family of generalized Catalan numbers that satisfy a(n;r)=r*a(n-1;r)+sum{k=1..n-2, a(k)*a(n-1-k;r)}, a(0;r)=a(1;r)=1. The Hankel transform of a(n;r) is h(n)=sum{k=0..n, T(n,k)*r^k} with g.f. (1-x)/(1-r*x+x^2). These sequences include A086246, A000108, A002212. (End)
From Wolfdieter Lang, Jun 11 2011: (Start)
The Riordan array ((1+x)/(1+x^2),x/(1+x^2)) with entries Phat(n,k)= ((-1)^(n-k))*T(n,k) and o.g.f. Phat(x,z)=(1+z)/(1-x*z+z^2) for the row polynomials Phat(n,x) is related to Chebyshev C and S polynomials as follows.
Phat(n,x) = (R(n+1,x)-R(n,x))/(x+2) = S(2*n,sqrt(2+x))
  with R(n,x)=C_n(x) in the Abramowitz and Stegun notation, p. 778, 22.5.11. See A049310 for the S polynomials. Proof from the o.g.f.s.
Recurrence for the row polynomials Phat(n,x):
  Phat(n,x) = x*Phat(n-1,x) - Phat(n-2,x) for n>=1; Phat(-1,x)=-1, Phat(0,x)=1.
The A-sequence for this Riordan array Phat (see the W. Lang link under A006232 for A- and Z-sequences for Riordan matrices) is given by 1, 0, -1, 0, -1, 0, -2, 0, -5,.., starting with 1 and interlacing the negated A000108 with zeros (o.g.f. 1/c(x^2) = 1-c(x^2)*x^2, with the o.g.f. c(x) of A000108).
The Z-sequence has o.g.f. sqrt((1-2*x)/(1+2*x)), and it is given by A063886(n)*(-1)^n.
The A-sequence of the Riordan array T(n,k) is identical with the one for the Riordan array Phat, and the Z-sequence is -A063886(n).
(End)
The row polynomials P(n,x) are the characteristic polynomials of the adjacency matrices of the graphs which look like P_n (n vertices (nodes), n-1 lines (edges)), but vertex no. 1 has a loop. - Wolfdieter Lang, Nov 17 2011
From Wolfdieter Lang, Dec 14 2013: (Start)
The zeros of P(n,x) are x(n,j) = -2*cos(2*Pi*j/(2*n+1)), j=1..n. From P(n,x) = (-1)^n*S(2*n,sqrt(2-x)) (see, e.g., the Lemma 6 of the W. Lang link).
The discriminants of the P-polynomials are given in A052750. (End)

			The triangle T(n,k) begins:
n\k  0   1   1   3    4    5    6    7    8    9  10  11  12  13 14 15 ...
0:   1
1:  -1   1
2:  -1  -1   1
3:   1  -2  -1   1
4:   1   2  -3  -1    1
5:  -1   3   3  -4   -1    1
6:  -1  -3   6   4   -5   -1    1
7:   1  -4  -6  10    5   -6   -1    1
8:   1   4 -10 -10   15    6   -7   -1    1
9:  -1   5  10 -20  -15   21    7   -8   -1    1
10: -1  -5  15  20  -35  -21   28    8   -9   -1   1
11:  1  -6 -15  35   35  -56  -28   36    9  -10  -1   1
12:  1   6 -21 -35   70   56  -84  -36   45   10 -11  -1   1
13: -1   7  21 -56  -70  126   84 -120  -45   55  11 -12  -1   1
14: -1  -7  28  56 -126 -126  210  120 -165  -55  66  12 -13  -1  1
15:  1  -8 -28  84  126 -252 -210  330  165 -220 -66  78  13 -14 -1  1
...  reformatted and extended - _Wolfdieter Lang_, Jul 31 2014.
---------------------------------------------------------------------------
From _Paul Barry_, May 21 2009: (Start)
Production matrix is
-1, 1,
-2, 0, 1,
-2, -1, 0, 1,
-4, 0, -1, 0, 1,
-6, -1, 0, -1, 0, 1,
-12, 0, -1, 0, -1, 0, 1,
-20, -2, 0, -1, 0, -1, 0, 1,
-40, 0, -2, 0, -1, 0, -1, 0, 1,
-70, -5, 0, -2, 0, -1, 0, -1, 0, 1 (End)
Row polynomials as first difference of S polynomials:
P(3,x) = S(3,x) - S(2,x) = (x^3 - 2*x) - (x^2 -1) = 1 - 2*x - x^2 +x^3.
Alternative triangle recurrence (see a comment above): T(6,2) = T(5,2) + T(5,1) = 3 + 3 = 6. T(6,3) = -T(5,3) + 0*T(5,1) = -(-4) = 4. - _Wolfdieter Lang_, Jul 31 2014

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available).

Hyeong-Kwan Ju, On the sequence generated by a certain type of matrices, Honam Math. J. 39, No. 4, 665-675 (2017), Theorem 2.16.
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; see Definition 1, Lemma 6 and Remark 4.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31 (formula 5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A066170, A046854, A057077 (first column).

Row sums: A010892(n+1); repeat(1,0,-1,-1,0,1). Alternating row sums: A061347(n+2); repeat(1,-2,1).

A130777 := proc(n,k): (-1)^binomial(n-k+1,2)*binomial(floor((n+k)/2),k) end: seq(seq(A130777(n,k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 08 2011

T[n_, k_] := (-1)^Binomial[n - k + 1, 2]*Binomial[Floor[(n + k)/2], k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2017, from Maple *)

@CachedFunction
def A130777(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = A130777(n-1,k) if n==1 else 0
    return A130777(n-1,k-1) - A130777(n-2,k) - h
for n in (0..9): [A130777(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Number triangle T(n,k) = (-1)^C(n-k+1,2)*C(floor((n+k)/2),k). - Paul Barry, May 21 2009
From Wolfdieter Lang, Jun 11 2011: (Start)
Row polynomials: P(n,x) = sum(k=0..n, T(n,k)*x^k) = R(2*n+1,sqrt(2+x)) / sqrt(2+x), with Chebyshev polynomials R with coefficients given in A127672 (scaled T-polynomials).
R(n,x) is called C_n(x) in Abramowitz and Stegun's handbook, p. 778, 22.5.11.
P(n,x) = S(n,x)-S(n-1,x), n>=0, S(-1,x)=0, with the Chebyshev S-polynomials (see the coefficient triangle A049310).
O.g.f. for row polynomials: P(x,z):= sum(n>=0, P(n,x)*z^n ) = (1-z)/(1-x*z+z^2).
  (from the o.g.f. for R(2*n+1,x), n>=0, computed from the o.g.f. for the R-polynomials (2-x*z)/(1-x*z+z^2) (see A127672))
Proof of the Chebyshev connection from the o.g.f. for Riordan array property of this triangle (see the P. Barry comment above).
For the A- and Z-sequences of this Riordan array see a comment above. (End)
abs(T(n,k)) = A046854(n,k) = abs(A066170(n,k)) T(n,n-k) = A108299(n,k); abs(T(n,n-k)) = A065941(n,k). - Johannes W. Meijer, Aug 08 2011
From Wolfdieter Lang, Jul 31 2014: (Start)
Similar to the triangles A157751, A244419 and A180070 one can give for the row polynomials P(n,x) besides the usual three term recurrence another one needing only one recurrence step. This uses also a negative argument, namely P(n,x) = (-1)^(n-1)*(-1 + x/2)*P(n-1,-x) + (x/2)*P(n-1,x), n >= 1, P(0,x) = 1. Proof by computing the o.g.f. and comparing with the known one. This entails the alternative triangle recurrence T(n,k) = (-1)^(n-k)*T(n-1,k) + (1/2)*(1 + (-1)^(n-k))*T(n-1,k-1), n >= m >= 1, T(n,k) = 0 if n < k and T(n,0) = (-1)^floor((n+1)/2) = A057077(n+1). [P(n,x) recurrence corrected Aug 03 2014]
(End)

New name and Chebyshev comments by Wolfdieter Lang, Jun 11 2010

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A034807 Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.

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A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.

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A005581 a(n) = (n-1)*n*(n+4)/6.

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A111125 Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.

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A053122 Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).

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A263916 Coefficients of the Faber partition polynomials.

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A005581 a(n) = (n-1)n(n+4)/6.

A111125 Triangle read by rows: T(k,s) = ((2k+1)/(2s+1))binomial(k+s,2s), 0 <= s <= k.

A127677 Scaled coefficient table for Chebyshev polynomials 2T(2n, sqrt(x)/2) (increasing even scaled powers, without zero entries).