A136493 Triangle of coefficients of characteristic polynomials of symmetrical pentadiagonal matrices of the type (1,-1,1,-1,1).

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

Offset: 0

Author

Roger L. Bagula, Mar 21 2008

Keywords

• tabl
• sign

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)

Examples

			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;
		

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

Crossrefs

Cf. A011658, A087509, A124805 ff., A124696 ff., A124999 ff., A161680, A181983.

Programs

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]]

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)

Extensions

Edited by Georg Fischer, Mar 29 2021