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%I #17 Aug 17 2023 08:17:37
%S 1,-1,1,1,-2,0,-1,3,0,0,1,-4,1,2,0,-1,5,-3,-5,1,1,1,-6,6,8,-5,-2,1,-1,
%T 7,-10,-10,14,4,-4,0,1,-8,15,10,-29,-4,12,0,0,-1,9,-21,-7,50,-4,-30,4,
%U 4,0,1,-10,28,0,-76,28,61,-20,-15,2,1
%N Triangle of coefficients of characteristic polynomials of symmetrical pentadiagonal matrices of the type (1,-1,1,-1,1).
%C From _Georg Fischer_, Mar 29 2021: (Start)
%C The pentadiagonal matrices have 1 in the main diagonal, -1 in the first lower and upper diagonal, 1 in the second lower and upper diagonal, and 0 otherwise.
%C The linear recurrences that yield A124805, A124806, A124807 and similar can be derived from the rows of this triangle (the first element of a row must be removed and multiplied onto the remaining elements).
%C This observation extends to other sequences. For example the linear recurrence signature (5,-6,2,4,0) of A124698 "Number of base 5 circular n-digit numbers with adjacent digits differing by 1 or less" can be derived from the coefficients of the characteristic polynomial of a tridiagonal (type -1,1,-1) 5 X 5 matrix.
%C (End)
%D Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.
%H G. C. Greubel, <a href="/A136493/b136493.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Sum_{k=1..n} T(n, k) = (-1)^(n mod 3) * A087509(n+1) + [n=1].
%F From _G. C. Greubel_, Aug 01 2023: (Start)
%F T(n, n) = A011658(n+2).
%F T(n, 1) = (-1)^(n-1).
%F T(n, 2) = A181983(n-1).
%F T(n, 3) = (-1)^(n-3)*A161680(n-3). (End)
%e Triangle begins:
%e 1;
%e -1, 1;
%e 1, -2, 0;
%e -1, 3, 0, 0;
%e 1, -4, 1, 2, 0;
%e -1, 5, -3, -5, 1, 1;
%e 1, -6, 6, 8, -5, -2, 1;
%e -1, 7, -10, -10, 14, 4, -4, 0;
%e 1, -8, 15, 10, -29, -4, 12, 0, 0;
%e -1, 9, -21, -7, 50, -4, -30, 4, 4, 0;
%e 1, -10, 28, 0, -76, 28, 61, -20, -15, 2, 1;
%t T[n_, m_]:= Piecewise[{{-1, 1+m==n || m==1+n}, {1, 2+m==n || m==n || m==2+n}}];
%t MO[d_]:= Table[T[n, m], {n,d}, {m,d}];
%t CL[n_]:= CoefficientList[CharacteristicPolynomial[MO[n], x], x];
%t Join[{{1}}, Table[Reverse[CL[n]], {n,10}]]//Flatten
%t (* For the signature of A124698 added by _Georg Fischer_, Mar 29 2021 : *)
%t Reverse[CoefficientList[CharacteristicPolynomial[{{1,-1,0,0,0}, {-1, 1,-1,0,0}, {0,-1,1,-1,0}, {0,0,-1,1,-1}, {0,0,0,-1,1}}, x], x]]
%Y Cf. A011658, A087509, A124805 ff., A124696 ff., A124999 ff., A161680, A181983.
%K tabl,sign
%O 0,5
%A _Roger L. Bagula_, Mar 21 2008
%E Edited by _Georg Fischer_, Mar 29 2021
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