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Search: id:A125090
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| A125090 |
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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,...). |
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+0
2
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| 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, -6, 9, 5, -15, 0, 5, 0, 1, -7, 14, 1, -25, 9, 12, -3, -1, 1, -8, 20, -7, -35, 29, 18, -15, -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
(list; table; graph; listen)
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OFFSET
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1,8
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COMMENT
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The characteristic polynomial of the n X n matrix has a root = 1+2*cos(2*Pi/(2n+1)).
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FORMULA
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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.
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EXAMPLE
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Triangle starts:
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, -6, 9, 5, -15, 0, 5, 0;
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MAPLE
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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
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CROSSREFS
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Cf. A104562.
Sequence in context: A126886 A105685 A110858 this_sequence A073266 A125692 A128258
Adjacent sequences: A125087 A125088 A125089 this_sequence A125091 A125092 A125093
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KEYWORD
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sign,tabl
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 19 2006
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EXTENSIONS
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Edited by njas, Nov 29 2006
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