A169593 Coefficients of characteristic polynomials of determinant equals trace matrices using Eulerian trace and factorial determinant.

-1, 1, -1, 1, -1, 1, 2, -1, 2, -1, 6, -4, 9, -6, 1, 24, -121, 264, -166, 24, -1, 120, -44616, 93340, -52950, 4345, -120, 1, 720, -296321796, 605003244, -321204409, 12686988, -164746, 720, -1, 5040, -49349521382400, 99624831647040, -51206316902496

Offset

0,7

Comments

Row sums are:

{-1, 0, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880,...}.

Traces are:

Table[Sum[M[n][[k, k]], {k, 1, n}], {n, 1, 10}]

{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880,...}

Determinants are:

Table[Det[M[n]], {n, 1, 10}]

{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880,...}

Links

Table of n, a(n) for n=0..39.

Example

{-1},
{1, -1},
{1, -1, 1},
{2, -1, 2, -1},
{6, -4, 9, -6, 1},
{24, -121, 264, -166, 24, -1},
{120, -44616, 93340, -52950, 4345, -120, 1},
{720, -296321796, 605003244, -321204409, 12686988, -164746, 720, -1},
{5040, -49349521382400, 99624831647040, -51206316902496, 936232732785, -5234439280, 8349390, -5040, 1},
{40320, -274297679317746753201, 550979304410032093440, -279071418382631643820, 2395643740989790080, -5853386029582998, 2935936463360, -550407180, 40320, -1},
{362880, -65390418299618584017607843840, 131052209744019041903924775936, -65933467896655077725833452000, 271979805136698554372590800, -303398293776695489224080, 45414063262861346088, -2365327872234750, 45644404725, -362880, 1}

Mathematica

Clear[M, n, m, k, a0, b]
(*Eulerian number expansion*)
p[t_] = (1 - x)/(1 - x*Exp[t*(1 - x)])
b = Table[ CoefficientList[ FullSimplify[ ExpandAll[(n!/ x)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 1, 10}]
(* using the Eulerian numbers down the main diagonal since their sum is a factorial*)
M[n_] := Table[If[ k == m && m < n, b[[n - 1]][[ k]], If[k == m + 1, 1, If[k == 1 && m == n, (-1)^(n + 1)*( n - 1)!, 0]]], {k, n}, {m, n}]
TableForm[Table[M[n], {n, 1, 10}]]
Table[Det[M[n]], {n, 1, 10}]
Table[Sum[M[n][[k, k]], {k, 1, n}], {n, 1, 10}]
a = Join[{{-1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 10}]]
Flatten[a]

Crossrefs

Sequence in context: A284465 A327899 A276157 * A057568 A220587 A195962

Adjacent sequences: A169590 A169591 A169592 * A169594 A169595 A169596

Keyword

sign,uned

Author

Roger L. Bagula, Dec 02 2009

Status

approved