A158391 A triangle of matrix polynomials:( dot version) M(d)=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]].Transpose[Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]].

1, 1, -1, 4, -6, 1, 36, -62, 20, -1, 576, -1064, 440, -50, 1, 14400, -27752, 13092, -2068, 105, -1, 518400, -1027872, 526240, -99568, 7476, -196, 1, 25401600, -51402528, 27827104, -5903560, 552368, -22488, 336, -1, 1625702400

Offset

0,4

Comments

Row sums are:

{1, 0, -1, -7, -97, -2224, -75519, -3547169, -219848959, -17364520592,

-1701503169057,...}.

Example matrix:

M(3)={{14, 5, 1},

{5, 5, 1},

{1, 1, 1}}

Links

Table of n, a(n) for n=0..36.

Formula

M(d)=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]].Transpose[Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]];

out_(n,m)=coefficients(characteristicpolynomial(M(n),x),x)

Example

{1},
{1, -1},
{4, -6, 1},
{36, -62, 20, -1},
{576, -1064, 440, -50, 1},
{14400, -27752, 13092, -2068, 105, -1},
{518400, -1027872, 526240, -99568, 7476, -196, 1},
{25401600, -51402528, 27827104, -5903560, 552368, -22488, 336, -1},
{1625702400, -3340564992, 1883221312, -432454176, 46632432, -2444400, 59004, -540, 1},
{131681894400, -273837169152, 159196654656, -38743828352, 4614308240, -285357704, 9115756, -139260, 825, -1},
{13168189440000, -27647080704000, 16465714101504, -4189435579520, 537035259392, -37331932704, 1435658576, -29713112, 302016, -1210, 1}

Mathematica

Clear[M, T, d, a, x, a0];
M[d_] :=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]].Transpose[Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]];
a0 = Table[M[d], {d, 1, 10}];
Table[Det[M[d]], {d, 1, 10}];
Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
Flatten[a]
Join[{1}, Table[Apply[Plus, CoefficientList[ Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}];

Crossrefs

Sequence in context: A333668 A193293 A258220 * A016491 A347052 A375366

Adjacent sequences: A158388 A158389 A158390 * A158392 A158393 A158394

Keyword

sign,tabl

Author

Roger L. Bagula and Gary W. Adamson, Mar 17 2009

Status

approved