A124037 - OEIS

%I #25 Jan 22 2020 02:20:19

%S 1,1,-1,2,-4,1,5,-13,7,-1,13,-40,33,-10,1,34,-120,132,-62,13,-1,89,

%T -354,483,-308,100,-16,1,233,-1031,1671,-1345,595,-147,19,-1,610,

%U -2972,5561,-5398,3030,-1020,203,-22,1,1597,-8495,17984,-20410,13893,-5943,1610,-268,25,-1,4181,-24110,56886,-73816,59059

%N Triangle read by rows: row n gives coefficients of increasing powers of x in characteristic polynomial of the matrix (-1)^n*M_n, where M_n is the tridiagonal matrix defined in the Comments line.

%C The matrices M_n for n=1, 2, 3, ... are:

%C 1 X 1 {{1}},

%C 2 X 2 {{1, -1}, {-1, 3}},

%C 3 X 3 {{1, -1, 0}, {-1, 3, -1}, {0, -1, 3}},

%C 4 X 4 {{1, -1, 0, 0}, {-1, 3, -1, 0}, {0, -1, 3, -1}, {0, 0, -1, 3}},

%C 5 X 5 {{1, -1, 0, 0, 0}, {-1, 3, -1, 0, 0}, {0, -1, 3, -1, 0}, {0, 0, -1, 3, -1}, {0, 0, 0, -1, 3}},

%C 6 X 6 {{1, -1, 0, 0, 0, 0}, {-1, 3, -1, 0, 0, 0}, {0, -1, 3, -1, 0, 0}, {0, 0, -1, 3, -1, 0}, { 0, 0, 0, -1, 3, -1}, {0, 0, 0, 0, -1, 3}}, ...

%C Subtriangle of the triangle given by (0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 04 2014

%C Riordan array ((1-2*x)/(1-3*x+x^2), -x/(1-3*x+x^2)). - _Philippe Deléham_, Mar 04 2014

%F T(n,k) = 3*T(n-1,k) - T(n-1,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 1, T(1,1) = -1, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Mar 04 2014

%F G.f.: -(-1+2*x)/(1-3*x+x^2+x*y). - _R. J. Mathar_, Aug 11 2015

%e Triangle begins:

%e {1},

%e {1, -1},

%e {2, -4, 1},

%e {5, -13, 7, -1},

%e {13, -40, 33, -10, 1},

%e {34, -120,132, -62, 13, -1},

%e {89, -354, 483, -308, 100, -16, 1},

%e For example, the characteristic polynomial of M_3 is x^3-7*x^2+13*x-5, so row 3 is 5, -13, 7, -1.

%e Triangle (0, 1, 1, 1, 0, 0, 0, ...) DELTA (1, -2, 0, 0, 0, ...) begins:

%e 1;

%e 0, 1;

%e 0, 1, -1;

%e 0, 2, -4, 1;

%e 0, 5, -13, 7, -1;

%e 0, 13, -40, 33, -10, 1;

%e 0, 34, -120, 132, -62, 13, -1;

%e 0, 89, -354, 483, -308, 100, -16, 1; - _Philippe Deléham_, Mar 04 2014

%t T[n_, m_, d_] := If[ n == m && n > 1 && m > 1, 3, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, 1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[ d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a]

%o (Sage)

%o @CachedFunction

%o def T(n,k):

%o     if n< 0: return 0

%o     if n==0: return 1 if k == 0 else 0

%o     h = T(n-1,k) if n==1 else 3*T(n-1,k)

%o     return T(n-1,k-1) - T(n-2,k) - h

%o A124037 = lambda n,k: (-1)^n*T(n,k)

%o for n in (0..9): [A124037(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012

%Y Cf. A126126.

%K sign,tabl

%O 0,4

%A _Gary W. Adamson_ and _Roger L. Bagula_, Nov 03 2006

%E Edited by _N. J. A. Sloane_, Mar 02 2008