A169593 - OEIS

%I #2 Mar 30 2012 17:34:35

%S -1,1,-1,1,-1,1,2,-1,2,-1,6,-4,9,-6,1,24,-121,264,-166,24,-1,120,

%T -44616,93340,-52950,4345,-120,1,720,-296321796,605003244,-321204409,

%U 12686988,-164746,720,-1,5040,-49349521382400,99624831647040,-51206316902496

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

%C Row sums are:

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

%C Traces are:

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

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

%C Determinants are:

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

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

%e {-1},

%e {1, -1},

%e {1, -1, 1},

%e {2, -1, 2, -1},

%e {6, -4, 9, -6, 1},

%e {24, -121, 264, -166, 24, -1},

%e {120, -44616, 93340, -52950, 4345, -120, 1},

%e {720, -296321796, 605003244, -321204409, 12686988, -164746, 720, -1},

%e {5040, -49349521382400, 99624831647040, -51206316902496, 936232732785, -5234439280, 8349390, -5040, 1},

%e {40320, -274297679317746753201, 550979304410032093440, -279071418382631643820, 2395643740989790080, -5853386029582998, 2935936463360, -550407180, 40320, -1},

%e {362880, -65390418299618584017607843840, 131052209744019041903924775936, -65933467896655077725833452000, 271979805136698554372590800, -303398293776695489224080, 45414063262861346088, -2365327872234750, 45644404725, -362880, 1}

%t Clear[M, n, m, k, a0, b]

%t (*Eulerian number expansion*)

%t p[t_] = (1 - x)/(1 - x*Exp[t*(1 - x)])

%t b = Table[ CoefficientList[ FullSimplify[ ExpandAll[(n!/ x)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 1, 10}]

%t (* using the Eulerian numbers down the main diagonal since their sum is a factorial*)

%t 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}]

%t TableForm[Table[M[n], {n, 1, 10}]]

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

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

%t a = Join[{{-1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 10}]]

%t Flatten[a]

%K sign,uned

%O 0,7

%A _Roger L. Bagula_, Dec 02 2009