A158390 A triangle of matrix polynomials: 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, -1, 1, -36, 9, 5, -1, 576, 64, -61, -5, 1, 14400, -1600, -1045, 105, 14, -1, -518400, -32400, 29088, 1593, -385, -14, 1, -25401600, 1587600, 1079200, -63481, -11521, 579, 30, -1, 1625702400, 65028096, -55637056, -2062912, 523001, 16045

Offset

0,4

Comments

Row sums are:

{1, 0, -4, -23, 575, 11873, -520517, -22809194, 1633567982, 122963894441,

-13226887049109,...}.

Example matrix:

M(3)={{1, 2, 3},

{2, 4, 0},

{3, 0, 0}}

Links

Table of n, a(n) for n=0..41.

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, -1, 1},
{-36, 9, 5, -1},
{576, 64, -61, -5, 1},
{14400, -1600, -1045, 105, 14, -1},
{-518400, -32400, 29088, 1593, -385, -14, 1},
{-25401600, 1587600, 1079200, -63481, -11521, 579, 30, -1},
{1625702400, 65028096, -55637056, -2062912, 523001, 16045, -1563, -30, 1},
{131681894400, -5267275776, -3616612416, 138811968, 28130697, -981153, -75500, 2167, 55, -1},
{-13168189440000, -365783040000, 301206312000, 7960416400, -2038537300, -48112905, 5258465, 99136, -4851, -55, 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: A174412 A177939 A209196 * A228836 A176419 A299471

Adjacent sequences: A158387 A158388 A158389 * A158391 A158392 A158393

Keyword

sign,tabl

Author

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

Status

approved