A124030 Binomial centered tridigonal matrices as a triangular sequence: t(n,m.d)=If[n + m - 1 == d, binomial[d - 1, n - 1], If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]].

1, 1, -1, 0, -2, 1, 0, -3, 4, -1, 3, -14, 19, -8, 1, 48, -173, 204, -89, 16, -1, 1505, -4866, 5173, -2082, 381, -32, 1, 108780, -325990, 316978, -113481, 18926, -1580, 64, -1, 19072536, -53887686, 48428411, -15201276, 2206536, -164222, 6469, -128, 1, 8332293760, -22465873081, 18859204368, -5176293234

Offset

1,5

Comments

These are pretty matrices in terms of symmetry. Matrices: 1 X 1 {{1}} 2 X 2 {{1, -1}, {-1, 1}} 3 X 3 {{1, -1, 0}, {-1, 2, -1}, {0, -1, 1}} 4 X 4 {{1, -1, 0, 0}, {-1, 3, -1, 0}, {0, -1, 3, -1}, {0, 0, -1, 1}} 5 X 5 {{1, -1, 0, 0, 0}, {-1, 4, -1, 0, 0}, {0, -1, 6, -1, 0}, {0, 0, -1, 4, -1}, {0, 0, 0, -1, 1}} 6 X 6 {{1, -1, 0, 0, 0, 0}, {-1, 5, -1, 0, 0, 0}, {0, -1, 10, -1, 0, 0}, {0, 0, -1, 10, -1, 0}, {0, 0, 0, -1, 5, -1}, {0, 0, 0, 0, -1, 1}}

Links

Table of n, a(n) for n=1..49.

Formula

t(n,m.d)=If[n + m - 1 == d, binomial[d - 1, n - 1], If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]]

Example

Triangular sequence:
{1},
{1, -1},
{0, 2, 1},
{0, 3, 2, -1},
{3, -4, -11, 2, 1},
{48, -13, -106, 21, 6, -1},
{-1505, 36, 2693, -58, -129, 2, 1},
{-108780, 5530, 171342, -8705, -5290, 268,20, -1}

Mathematica

An[d_] := Table[If[n + m - 1 == d, Binomial[d - 1, n - 1], If[n + m ==d, -1, If[n + m - 2 == d, -1, 0]]], {n, 1, d}, {m, 1, d}]; Join[An[1], Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]

Crossrefs

Sequence in context: A363154 A101603 A228161 * A166040 A106378 A094301

Adjacent sequences: A124027 A124028 A124029 * A124031 A124032 A124033

Keyword

uned,sign

Author

Roger L. Bagula, Nov 01 2006

Status

approved