A140882 Triangle by rows with row n formed by coefficients of the characteristic polynomial of the n X n tridiagonal matrix with m_{i,i} = 2 for i=1..n, m_{i,i-1} = m_{i,i+1} = -1 for i=2..n-1, and m_{1,2} = m_{n,n-1} = -2.

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

Offset

0,2

References

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

Links

Table of n, a(n) for n=0..65.

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

Cf. A228785, A267633.

Cf. A049310, A133156.

Sequence in context: A185740 A139360 A326759 * A334044 A143724 A143425

Adjacent sequences: A140879 A140880 A140881 * A140883 A140884 A140885

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