A124039 Tridiagonal matrices with zero center upper 3 as a triangular sequence: m(n,m,d)=If[ n == m && n > 1 && m > 1, 0, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, 3, 0]]].

3, 3, -1, -1, -3, 1, -3, 2, 3, -1, 1, 6, -3, -3, 1, 3, -3, -9, 4, 3, -1, -1, -9, 6, 12, -5, -3, 1, -3, 4, 18, -10, -15, 6, 3, -1, 1, 12, -10, -30, 15, 18, -7, -3, 1, 3, -5, -30, 20, 45, -21, -21, 8, 3, -1, -1, -15, 15, 60, -35, -63, 28, 24, -9, -3, 1

Offset

1,1

Comments

Matrices modeled on: {{k, -1, 0}, {-1, 0, -1}, {0, -1, 0}} K=1 is Steinbach: A066170 These are equivalent to the central variable determinant sequences: {{y-k, -1, 0}, {-1, y, -1}, {0, -1, y}}

Links

Table of n, a(n) for n=1..66.

Formula

m(n,m,d)=If[ n == m && n > 1 && m > 1, 0, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, 3, 0]]]

Example

Triangle begins:
{{3}},--> Det[{{3}}]=3
{3, -1},
{-1, -3, 1},
{-3, 2, 3, -1},
{1, 6, -3, -3, 1},
{3, -3, -9, 4,3, -1},
{-1, -9, 6, 12, -5, -3, 1}

Mathematica

T[n_, m_, d_] := If[ n == m && n > 1 && m > 1, 0, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, 3, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[ d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a]

Prog

(Sage)
@CachedFunction
def A124039(n, k):  #  With T(0, 0) = 1!
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 3*A124039(n-1, k) if n==1 else 0
    return A124039(n-1, k-1) - A124039(n-2, k) - h
for n in (0..9): [A124039(n, k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Crossrefs

Cf. A066170.

Sequence in context: A303992 A365213 A126717 * A350617 A337743 A335624

Adjacent sequences: A124036 A124037 A124038 * A124040 A124041 A124042

Keyword

uned,tabl,sign

Author

Roger L. Bagula and Gary W. Adamson, Nov 03 2006

Status

approved