OFFSET

0,2

REFERENCES

Kemeny, Snell and Thompson, Introduction to Finite Mathematics, 1966, Prentice-Hall, New Jersey, Section 3, Chapter VII, page 407.

LINKS

Pentti Haukkanen, Jorma Merikoski, and Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.

FORMULA

Starting with the third row (0, -4, 1), these coefficients appear to occur in the odd row polynomials in inverse powers of u for the o.g.f. (u - 4)/(u - u*x + x^2) of A267633, which generates a Fibonacci-type running average of adjacent polynomials of the o.g.f. involving these polynomials and others embedded in A228785. - Tom Copeland, Jan 16 2016

A generating function for the third and later shifted, unsigned rows is (4 + t) / (1 - (2 + t)*x + x^2) = Sum_{n >= 0} (4+t)*U_n((2 + t)/2)*x^n = 4 + t + (8 + 6*t + t^2)*x + ..., where U_n(t) are the Chebyshev polynomials of the second kind of A133156. U_n((2 + t)/2) = V_n((2 + t)), where V_n(t) are the Chebyshev polynomials of A049310. A formula for the coefficients is given by Eqn. 8 in Haukkanen et al. - Tom Copeland, Apr 26 2024

EXAMPLE

1;

2, -1;

0, -4, 1;

0, -8, 6, -1;

0, -12, 19, -8, 1;

0, -16, 44, -34, 10, -1;

0, -20, 85, -104, 53, -12, 1;

0, -24, 146, -259, 200, -76, 14, -1;

0, -28, 231, -560, 606, -340, 103, -16, 1;

0, -32, 344, -1092, 1572, -1210, 532, -134, 18, -1;

0, -36, 489, -1968, 3630, -3652, 2171, -784, 169, -20, 1;

...

MAPLE

# Assume T(1, 0) = 0 instead of 2.

# Then a slightly modified form of Copeland's second comment gives

# T(n, k) = [t^k] [x^n] gf where

gf := ((t - 4)*t*x^2) / ((t - 2)*x + x^2 + 1) - t*x + 1:

ser := series(gf, x, 12): cx := n -> coeff(ser, x, n):

for n from 0 to 10 do lprint(seq(coeff(cx(n), t, k), k = 0..n)) od;

# Peter Luschny, Apr 27 2024

MATHEMATICA

T[n_, m_, d_] := If[ n == m, 2, If[(n == d && m == d - 1) || ( n == 1 && m == 2), -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[{{1}}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 1, 10}]];

Flatten[a]

CROSSREFS

KEYWORD

tabl,sign

AUTHOR

Roger L. Bagula and Gary W. Adamson, Jul 22 2008

EXTENSIONS

Edited by the editors of the OEIS, Apr 27 2024

STATUS

approved