This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A140882 #40 Apr 28 2024 03:01:28
%S A140882 1,2,-1,0,-4,1,0,-8,6,-1,0,-12,19,-8,1,0,-16,44,-34,10,-1,0,-20,85,
%T A140882 -104,53,-12,1,0,-24,146,-259,200,-76,14,-1,0,-28,231,-560,606,-340,
%U A140882 103,-16,1,0,-32,344,-1092,1572,-1210,532,-134,18,-1,0,-36,489,-1968,3630,-3652,2171,-784,169,-20,1
%N 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.
%D A140882 Kemeny, Snell and Thompson, Introduction to Finite Mathematics, 1966, Prentice-Hall, New Jersey, Section 3, Chapter VII, page 407.
%H A140882 Pentti Haukkanen, Jorma Merikoski, and Seppo Mustonen, <a href="https://acta.sapientia.ro/en/series/mathematica/publications-acta-math/mathematica-contents-of-volume-6-number-2-2014/some-polynomials-associated-with-regular-polygons">Some polynomials associated with regular polygons</a>, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
%F A140882 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
%F A140882 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
%e A140882 1;
%e A140882 2, -1;
%e A140882 0, -4, 1;
%e A140882 0, -8, 6, -1;
%e A140882 0, -12, 19, -8, 1;
%e A140882 0, -16, 44, -34, 10, -1;
%e A140882 0, -20, 85, -104, 53, -12, 1;
%e A140882 0, -24, 146, -259, 200, -76, 14, -1;
%e A140882 0, -28, 231, -560, 606, -340, 103, -16, 1;
%e A140882 0, -32, 344, -1092, 1572, -1210, 532, -134, 18, -1;
%e A140882 0, -36, 489, -1968, 3630, -3652, 2171, -784, 169, -20, 1;
%e A140882 ...
%p A140882 # Assume T(1, 0) = 0 instead of 2.
%p A140882 # Then a slightly modified form of Copeland's second comment gives
%p A140882 # T(n, k) = [t^k] [x^n] gf where
%p A140882 gf := ((t - 4)*t*x^2) / ((t - 2)*x + x^2 + 1) - t*x + 1:
%p A140882 ser := series(gf, x, 12): cx := n -> coeff(ser, x, n):
%p A140882 for n from 0 to 10 do lprint(seq(coeff(cx(n), t, k), k = 0..n)) od;
%p A140882 # _Peter Luschny_, Apr 27 2024
%t A140882 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]]];
%t A140882 M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];
%t A140882 a = Join[{{1}}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 1, 10}]];
%t A140882 Flatten[a]
%Y A140882 Cf. A228785, A267633.
%Y A140882 Cf. A049310, A133156.
%K A140882 tabl,sign
%O A140882 0,2
%A A140882 _Roger L. Bagula_ and _Gary W. Adamson_, Jul 22 2008
%E A140882 Edited by the editors of the OEIS, Apr 27 2024