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A104562
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Inverse of the Motzkin triangle A064189.
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5
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1, -1, 1, 0, -2, 1, 1, 1, -3, 1, -1, 2, 3, -4, 1, 0, -4, 2, 6, -5, 1, 1, 2, -9, 0, 10, -6, 1, -1, 3, 9, -15, -5, 15, -7, 1, 0, -6, 3, 24, -20, -14, 21, -8, 1, 1, 3, -18, -6, 49, -21, -28, 28, -9, 1, -1, 4, 18, -36, -35, 84, -14, -48, 36, -10, 1, 0, -8, 4, 60, -50, -98, 126, 6, -75, 45, -11, 1, 1, 4, -30, -20, 145, -36, -210, 168, 45, -110, 55
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Or, triangle read by rows: T(0,0)=1; for n>=1 T(n,k) is the coefficient of x^k in the monic characteristic polynomial of the n X n tridiagonal matrix with 1's on the main, sub- and superdiagonal (0<=k<=n). The characteristic polynomial has a root 1+2cos(Pi/(n+1)). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 19 2006
Row sums have g.f. 1/(1+x^2); diagonal sums are (-1)^n. Riordan array (1/(1+x+x^2), x/(1+x+x^2)).
Apart from signs, identical to A101950.
Or, triangle read by rows in which row n gives coefficients of characteristic polynomial of tridiagonal matrix with 1's on the main diagonal and -1's on the two adjacent diagonals. For example: M(3)={{1, -1, 0}, {-1, 1, -1}, {0, -1, 1}}. - From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 15 2008
Subtriangle of the triangle given by [0,-1,1,-1,0,0,0,0,0,0,0,...) DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 27 2010]
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REFERENCES
| Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.
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FORMULA
| Number triangle T(n, k)=sum{j=0..n, (-1)^(k-j)*(-1)^((n-j)/2) C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2}
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EXAMPLE
| Triangle starts:
1;
-1,1;
0,-2,1;
1,1,-3,1;
-1,2,3,-4,1;
0,-4,2,6,-5,1;
Triangle [0,-1,1,-1,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] begins : 1 ; 0,1 ; 0,-1,1 ; 0,0,-2,1 ; 0,1,1,-3,1 ; 0,-1,2,3,-4,1 ; ... [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 27 2010]
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MAPLE
| with(linalg): m:=proc(i, j) if abs(i-j)<=1 then 1 else 0 fi end: T:=(n, k)->coeff(charpoly(matrix(n, n, m), x), x, k): 1; for n from 1 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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MATHEMATICA
| a0[n_] := 1; b[n_] := -1; T[n_, m_, d_] := If[ n == m, a0[n], If[n == m - 1 || n == m + 1, If[n == m - 1, b[m - 1], If[n == m + 1, b[n - 1], 0]], 0]]; MO[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[ MO[n], x], x], {n, 1, 10}]]; Flatten[a] - More terms from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 15 2008
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CROSSREFS
| Cf. A125090, A101950.
Sequence in context: A135222 A124094 A101950 * A164306 A111603 A180178
Adjacent sequences: A104559 A104560 A104561 * A104563 A104564 A104565
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KEYWORD
| easy,sign,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Mar 15 2005
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 10 2008
Typo correction in the Roger L. Bagula comment and Mathematica section by Wolfdieter Lang, Nov 22 2011.
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