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%I #10 Aug 09 2015 01:14:20
%S 1,-2,1,-1,-2,1,2,-2,-2,1,1,4,-3,-2,1,-2,3,6,-4,-2,1,-1,-6,6,8,-5,-2,
%T 1,2,-4,-12,10,10,-6,-2,1,1,8,-10,-20,15,12,-7,-2,1,-2,5,20,-20,-30,
%U 21,14,-8,-2,1,-1,-10,15,40,-35,-42,28,16,-9,-2,1
%N Determinants of tridiagonal matrices in y with upper diagonal y-2: m(n,n,d)=If[ n == m && n > 1 && m > 1, y, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, y - 2, 0]]] Det(m,n,m,d)=P(d,y).
%C Matrices modeled on: {{-2 + y, -1, 0}, {-1, y, -1}, {0, -1, y}} The upper y-1 gives the Steinbach polynomials A066170.
%F m(n,n,d)=If[ n == m && n > 1 && m > 1, y, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, y - 2, 0]]]; Det(m,n,m,d)=P(d,y)
%e Triangular sequence:
%e {1},
%e {-2, 1},
%e {-1, -2, 1},
%e {2, -2, -2, 1},
%e {1, 4, -3, -2, 1},
%e {-2, 3, 6, -4, -2, 1},
%e {-1, -6, 6, 8, -5, -2, 1},
%e {2, -4, -12,10, 10, -6, -2, 1},
%e {1, 8, -10, -20, 15, 12, -7, -2, 1},
%e {-2, 5, 20, -20, -30, 21, 14, -8, -2, 1},
%e {-1, -10, 15, 40, -35, -42, 28, 16, -9, -2, 1}
%t T[n_, m_, d_] := If[ n == m && n >1 && m > 1, y, If[n == m - 1 || n == m + 1, -1,If[n == m == 1, y - 2, 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}] a = Join[{{1}}, Table[CoefficientList[Table[Det[M[d]], {d, 1, 10}][[d]], y], {d, 1, 10}]]; Flatten[a]
%o (Sage)
%o @CachedFunction
%o def A124038(n,k):
%o if n< 0: return 0
%o if n==0: return 1 if k == 0 else 0
%o h = 2*A124038(n-1,k) if n==1 else 0
%o return A124038(n-1,k-1) - A124038(n-2,k) - h
%o for n in (0..9): [A124038(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012
%Y Cf. A066170.
%K uned,sign
%O 1,2
%A _Gary W. Adamson_ and _Roger L. Bagula_, Nov 03 2006
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