A124038 Determinants of tridiagonal matrices in y with upper diagonal y-2: m(n,n,d)=If[ n == m && n > 1 && m > 1, y, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, y - 2, 0]]] Det(m,n,m,d)=P(d,y).

1, -2, 1, -1, -2, 1, 2, -2, -2, 1, 1, 4, -3, -2, 1, -2, 3, 6, -4, -2, 1, -1, -6, 6, 8, -5, -2, 1, 2, -4, -12, 10, 10, -6, -2, 1, 1, 8, -10, -20, 15, 12, -7, -2, 1, -2, 5, 20, -20, -30, 21, 14, -8, -2, 1, -1, -10, 15, 40, -35, -42, 28, 16, -9, -2, 1

Offset

1,2

Comments

Matrices modeled on: {{-2 + y, -1, 0}, {-1, y, -1}, {0, -1, y}} The upper y-1 gives the Steinbach polynomials A066170.

Links

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

Formula

m(n,n,d)=If[ n == m && n > 1 && m > 1, y, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, y - 2, 0]]]; Det(m,n,m,d)=P(d,y)

Example

Triangular sequence:
{1},
{-2, 1},
{-1, -2, 1},
{2, -2, -2, 1},
{1, 4, -3, -2, 1},
{-2, 3, 6, -4, -2, 1},
{-1, -6, 6, 8, -5, -2, 1},
{2, -4, -12,10, 10, -6, -2, 1},
{1, 8, -10, -20, 15, 12, -7, -2, 1},
{-2, 5, 20, -20, -30, 21, 14, -8, -2, 1},
{-1, -10, 15, 40, -35, -42, 28, 16, -9, -2, 1}

Mathematica

T[n_, m_, d_] := If[ n == m && n >1 && m > 1, y, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, y - 2, 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}] a = Join[{{1}}, Table[CoefficientList[Table[Det[M[d]], {d, 1, 10}][[d]], y], {d, 1, 10}]]; Flatten[a]

Prog

(Sage)
@CachedFunction
def A124038(n, k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A124038(n-1, k) if n==1 else 0
    return A124038(n-1, k-1) - A124038(n-2, k) - h
for n in (0..9): [A124038(n, k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Crossrefs

Cf. A066170.

Sequence in context: A137672 A141272 A281527 * A029311 A217869 A116674

Adjacent sequences: A124035 A124036 A124037 * A124039 A124040 A124041

Keyword

uned,sign

Author

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

Status

approved