cp's OEIS Frontend

A125090 Triangle read by rows: T(0,0)=1; for 0<=k<=n, n>=1, T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the tridiagonal n X n matrix with diagonal (0,1,1,...) and super- and subdiagonals (1,1,1,...).

%I A125090 #11 Jun 13 2019 09:37:40
%S A125090 1,1,0,1,-1,-1,1,-2,-1,1,1,-3,0,3,0,1,-4,2,5,-2,-1,1,-5,5,6,-7,-2,1,1,
%T A125090 -6,9,5,-15,0,5,0,1,-7,14,1,-25,9,12,-3,-1,1,-8,20,-7,-35,29,18,-15,
%U A125090 -3,1,1,-9,27,-20,-42,63,14,-42,0,7,0,1,-10,35,-39,-42,112,-14,-85,24,22,-4,-1,1,-11,44,-65,-30,174,-84,-134,95,40,-26,-4
%N A125090 Triangle read by rows: T(0,0)=1; for 0<=k<=n, n>=1, T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the tridiagonal n X n matrix with diagonal (0,1,1,...) and super- and subdiagonals (1,1,1,...).
%C A125090 The characteristic polynomial of the n X n matrix has a root = 1+2*cos(2*Pi/(2n+1)).
%F A125090 f(n,x)=(x-1)f(n-1,x)-f(n-2,x), where f(n,x) is the monic characteristic polynomial of the n X n matrix from the definition and f(0,x)=1.
%e A125090 Triangle starts:
%e A125090 1;
%e A125090 1, 0;
%e A125090 1, -1, -1;
%e A125090 1, -2, -1, 1;
%e A125090 1, -3, 0, 3, 0;
%e A125090 1, -4, 2, 5, -2, -1;
%e A125090 1, -5, 5, 6, -7, -2, 1;
%e A125090 1, -6, 9, 5, -15, 0, 5, 0;
%p A125090 with(linalg): m:=proc(i,j): if i=1 and j=1 then 0 elif i=j then 1 elif abs(i-j)=1 then 1 else 0 fi end: T:=proc(n,k) if n=0 and k=0 then 1 else coeff(charpoly(matrix(n,n,m),x),x,n-k) fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A125090 T[n_, k_] := T[n, k] = Which[n < 0, 0, n == 0, If[k == 0, 1, 0], True, T[n-1, k-1] - T[n-2, k] - If[n == 1, 0, T[n-1, k]]];
%t A125090 A[n_, k_] := T[n, n-k];
%t A125090 Table[A[n, k], {n, 0, 12}, {k, 0, n}] (* _Jean-François Alcover_, Jun 13 2019, after _Peter Luschny_ *)
%o A125090 (Sage)
%o A125090 @CachedFunction
%o A125090 def T(n, k):
%o A125090     if n< 0: return 0
%o A125090     if n==0: return 1 if k == 0 else 0
%o A125090     h = 0 if n==1 else T(n-1, k)
%o A125090     return T(n-1,k-1) - T(n-2, k) - h
%o A125090 A125090 = lambda n,k: T(n, n-k)
%o A125090 for n in (0..9): [A125090(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012
%Y A125090 Cf. A104562.
%K A125090 sign,tabl
%O A125090 1,8
%A A125090 _Gary W. Adamson_, Nov 19 2006
%E A125090 Edited by _N. J. A. Sloane_, Nov 29 2006