A168242 Coefficient triangle sequence of characteristic polynomials of a Fermat like matrix:M(n)=Eulerian n-th matrix: F(n)=M(n).Transpose[M(n)]]

1, 1, -1, 1, -3, 1, 1, -29, 21, -1, 1, -1769, 1872, -265, 1, 1, -615871, 1227274, -405530, 5975, -1, 1, -1124878187, 14488841796, -8196098025, 250869024, -194883, 1, 1, -10146337387971, 4127467195814962, -2684293284374908

Offset

0,5

Comments

Row Sums are:

{1, 0, -1, -8, -160, 211848,5418539727, 1568832578459224, 5586114023994799591396,

-416657044755533539390036118560, -14128568453639379750002082917435886133265, -6849588404991830798534151932598866046674668061916984...}.

Example matrix F(3);

{{3, 5, 1},

{5, 17, 4},

{1, 4, 1}}

References

L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 172

Links

Table of n, a(n) for n=0..31.

Example

{1},
{1, -1},
{1, -3, 1},
{1, -29, 21, -1},
{1, -1769, 1872, -265, 1},
{1, -615871, 1227274, -405530, 5975, -1},
{1, -1124878187, 14488841796, -8196098025, 250869024, -194883, 1},
{1, -10146337387971, 4127467195814962, -2684293284374908, 136124480159052, -319484649614, 8897703, -1},
{1, -428247824076704947, 17914574195515702665641, -13896536495233638792866, 1577809812703562226910, -9306019885601872094, 778719385854493, -533785743, 1},
{1, -81122352002183205692001, 1013200645627301010738899491371, -2171326301305554075331374889621, 748623927505727977255573077308, -7156540983035789703544116336, 1305525584958241674283545, -3205618448985143519, 40926870693, -1},
{1, -66885549293275814462767924797, 700501791981375381769184102256200146631, -28023249349569184787979186261051800365390, 13350702011670792795258543143392523163311, -156586293448931879782269330892154553133, 63386232099122631948920609947972715, -438644853880761289559728814794, 21273733616165855220783, -3893960978593, 1},
{1, -234732980513118576078558469331193281, 5621271025686372941421637129418789177249005788203, -13305553109795291932551487280535074672188567064406267, 6564379514567042192853283682636347534279562461093950, -114178043014292921637849211194413456481495109216258, 141964308173916470071468540296775755854414852854, -2083148469963100139599682004755602288051786, 278086488638019757478602360279335075, -214476940412956125417947983, 450564987828509, -1}

Mathematica

Clear[t, T, M, F];
t[n_, k_] = Sum[(-1)^j Binomial[n, j](k + 1 - j)^(n - 1), {j, 0, k}];
T[n_, k_] := If[n >= m, t[n + 2, k], 0];
M[n_] := Table[T[m, k], {k, 0, n}, {m, 0, n}];
F[n_] := M[n].Transpose[M[n]];
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[F[n], x], x], {n, 0, 10}]];
Flatten[%]

Crossrefs

Sequence in context: A277170 A216922 A245243 * A332540 A215750 A126465

Adjacent sequences: A168239 A168240 A168241 * A168243 A168244 A168245

Keyword

sign,uned

Author

Roger L. Bagula, Nov 21 2009

Status

approved