OFFSET
0,4
COMMENTS
The length of row n is A008619(n).
Essentially equals a signed version of A034807, the triangle of Lucas polynomials. The initial n coefficients of 1/C(x)^n consist of row n followed by floor((n-1)/2) zeros for n > 0.
For the following formula for 1/C(x)^n see the W. Lang reference, proposition 1 on p. 411:
1/C(x)^n = (sqrt(x))^n*(S(n,1/sqrt(x)) - sqrt(x)*S(n-1,1/sqrt(x))*C(x)), n >= 0, with the Chebyshev polynomials S(n,x) with coefficients given in A049310. See also the coefficient array A115139 for P(n,x) = (sqrt(x)^(n-1))*S(n-1, 1/sqrt(x)). - Wolfdieter Lang, Sep 14 2013
This triangular array is composed of interleaved rows of reversed, A127677 (cf. A156308, A217476, A263916) and reversed, signed A111125. - Tom Copeland, Nov 07 2015
It seems that the n-th row lists the coefficients of the HOMFLYPT (HOMFLY) polynomial reduced to one variable for link family n, see Jablan's slide 38. - Andrey Zabolotskiy, Jan 16 2018
For n >= 1 row n gives the coefficients of the Girard-Waring formula for the sum of x1^n + x2^n in terms of the elementary symmetric functions e_1(x1,x2) = x1 + x2 and e_2(x1,x2) = x1*x2. This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions with parts larger than 2 eliminated. E.g., n = 4: x1^4 + x2^4 = 1*e1^4 - 4*e1^3*e2 + 2*e1*e2^2. See also A115131, row n = 4, with the mentioned partitions omitted. - Wolfdieter Lang, May 03 2019
Row n lists the coefficients of the n-th Faber polynomial for the replicable function given in A154272 with offset -1. - Ben Toomey, May 12 2020
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 200, flattened)
Tom Copeland, Addendum to Elliptic Lie Triad
G. Dattoli, E. Di Palma and E. Sabia, Cardan Polynomials, Chebyshev Exponents, Ultra-Radicals and Generalized Imaginary Units, Advances in Applied Clifford Algebras, 2014.
Pentti Haukkanen, Jorma Merikoski and Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
S. Jablan, Knots, computers, conjectures
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq. (23) with n -> -n and eq. (20).
Jorma K. Merikoski, Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials, Notes on Num. Theor. and Disc. Math. (2021) Vol. 27, No. 2, 79-87.
FORMULA
T(n,k) = (-1)^k*( C(n-k,k) + C(n-k-1,k-1) ) for n >= 0, 0 <= k <= floor(n/2).
T(0,0) = 1; T(n,k) = (-1)^k*n*binomial(n-k,k)/(n-k), k = 0..floor(n/2). - Wolfdieter Lang, May 03 2019
EXAMPLE
The irregular triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 ...
-------------------------------------------------
0: 1
1: 1
2: 1 -2
3: 1 -3
4: 1 -4 2
5: 1 -5 5
6: 1 -6 9 -2
7: 1 -7 14 -7
8: 1 -8 20 -16 2
9: 1 -9 27 -30 9
10: 1 -10 35 -50 25 -2
11: 1 -11 44 -77 55 -11
12: 1 -12 54 -112 105 -36 2
13: 1 -13 65 -156 182 -91 13
14: 1 -14 77 -210 294 -196 49 -2
... (reformatted - Wolfdieter Lang, May 03 2019)
MATHEMATICA
T[0, 0] = 1; T[n_, k_] := (-1)^k (Binomial[n-k, k] + Binomial[n-k-1, k-1]);
Table[T[n, k], {n, 0, 14}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Jun 04 2018 *)
PROG
(PARI) {T(n, k)=if(k>n\2, 0, (-1)^k*(binomial(n-k, k)+binomial(n-k-1, k-1)))}
CROSSREFS
KEYWORD
sign,easy,tabf
AUTHOR
Paul D. Hanna, Aug 21 2007
STATUS
approved