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A140882
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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.
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2
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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
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OFFSET
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0,2
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REFERENCES
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Kemeny, Snell and Thompson, Introduction to Finite Mathematics, 1966, Prentice-Hall, New Jersey, Section 3, Chapter VII, page 407.
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LINKS
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FORMULA
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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
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EXAMPLE
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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;
...
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MAPLE
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# 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;
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Edited by the editors of the OEIS, Apr 27 2024
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STATUS
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approved
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