A158419 A triangle sequence of matrix polynomials: M(d)=Table[If[ m == n, n!, 1], {n, 1, d}, {m, 1, d}].Table[If[ m == n, d!/n!, -1], {n, 1, d}, {m, 1, d}].

1, 1, -1, 1, -2, 1, 30, -36, 12, -1, 68425, -22088, 2250, -84, 1, 6644998080, -462755822, 11295304, -121305, 580, -1, 42023309490821085, -532293793565874, 2536970688039, -5906096588, 7149243, -4290, 1, 26556532700608740463107950

Offset

0,5

Comments

Row sums are:

1, 0, 0, 5, 48504, 6193416836, 41493546768991616, 26504024825880784909569216,

2332709250270908106278491173069190784,

37399170617171570318305229620679111291900091279680,

138274453599768944166991999951900168216586552198815075762411849728...

Example matrix is:

M(3)={{4, 1, -1},

{3, 4, -2},

{-1, -4, 4}}.

This result is a product of matrices designed so that the diagonal

comes out the same integer.

Matrix diagonal sequence is:

{1, 1, 4, 21, 116, 715, 5034, 40313, 362872, 3628791,...}.

Links

Table of n, a(n) for n=0..28.

Formula

M(d)=Table[If[ m == n, n!, 1], {n, 1, d}, {m, 1, d}].Table[If[ m == n, d!/n!, -1], {n, 1, d}, {m, 1, d}];

out_(n,m)=coefficient(characteristicpolynomial(M(n),x),x).

Example

{1},
{1, -1},
{1, -2, 1},
{30, -36, 12, -1},
{68425, -22088, 2250, -84, 1},
{6644998080, -462755822, 11295304, -121305, 580, -1}, {42023309490821085, -532293793565874, 2536970688039, -5906096588, 7149243, -4290, 1},
{26556532700608740463107950, -52548930606504995256280, 41072580544021312950, -16705847783740933, 3853710807814, -506697522, 35238, -1},
{2333338806718406143341521201155376145, -629625963183454975722854581030348, 69519850167563258978622943400, -4164898983390708038987712, 149336122107141599374, -3297595134733164, 43904345592, -322504, 1},
{37400388617769706876059227784782872772071572259500, -1218017270285115982753099698893597916737879850, 16672276116958734715591544675368220325840, -127559335723136543998161968361745933, 605745216453559222052354128236, -1859909675898417187846209, 3702285323440407424, -4613211635175, 3265848, -1},
{138274940349898430483193483320560346218806349116410323550522943445, -486750864928700952008731075173338809176170535781335756167798, 735443019884273758279989766754250412248717315425241261, -635134097896528257062303632963480269576361349560, 349500791838986436397296951057896698761210, -128576386405874397174558733725642324, 32108244286792237055932681522, -5382397230460170947544, 580049882617425, -36287910, 1}

Mathematica

Clear[M, T, d, a, x, a0];
M[d_] := Table[If[ m == n, n!, 1], {n, 1, d}, {m, 1, d}].Table[If[ m == n, d!/n!, -1], {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}];
a = Join[{{1}}, Table[CoefficientList[Expand[ CharacteristicPolynomial[M[n], x]], x], {n, 1, 10}]];
Flatten[a]
Join[{1}, Table[Apply[Plus, CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}]];

Crossrefs

Sequence in context: A351710 A321442 A005693 * A074011 A054235 A016547

Adjacent sequences: A158416 A158417 A158418 * A158420 A158421 A158422

Keyword

sign,tabl

Author

Roger L. Bagula and Gary W. Adamson, Mar 18 2009

Status

approved