OFFSET
0,5
COMMENTS
From Georg Fischer, Mar 29 2021: (Start)
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.
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).
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.
(End)
REFERENCES
Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Sum_{k=1..n} T(n, k) = (-1)^(n mod 3) * A087509(n+1) + [n=1].
From G. C. Greubel, Aug 01 2023: (Start)
T(n, n) = A011658(n+2).
T(n, 1) = (-1)^(n-1).
T(n, 2) = A181983(n-1).
T(n, 3) = (-1)^(n-3)*A161680(n-3). (End)
EXAMPLE
Triangle begins:
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, 7, -10, -10, 14, 4, -4, 0;
1, -8, 15, 10, -29, -4, 12, 0, 0;
-1, 9, -21, -7, 50, -4, -30, 4, 4, 0;
1, -10, 28, 0, -76, 28, 61, -20, -15, 2, 1;
MATHEMATICA
T[n_, m_]:= Piecewise[{{-1, 1+m==n || m==1+n}, {1, 2+m==n || m==n || m==2+n}}];
MO[d_]:= Table[T[n, m], {n, d}, {m, d}];
CL[n_]:= CoefficientList[CharacteristicPolynomial[MO[n], x], x];
Join[{{1}}, Table[Reverse[CL[n]], {n, 10}]]//Flatten
(* For the signature of A124698 added by Georg Fischer, Mar 29 2021 : *)
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]]
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Roger L. Bagula, Mar 21 2008
EXTENSIONS
Edited by Georg Fischer, Mar 29 2021
STATUS
approved