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A136674
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Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial p(n,x) with p(0,x)=1, p(1,x) = 2-x, p(2,x) = 1-4x+x^2 and p(n,x) = (2-x)*p(n-1,x)-p(n-2,x) if n>2.
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9
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1, 2, -1, 1, -4, 1, 0, -8, 6, -1, -1, -12, 19, -8, 1, -2, -15, 44, -34, 10, -1, -3, -16, 84, -104, 53, -12, 1, -4, -14, 140, -258, 200, -76, 14, -1, -5, -8, 210, -552, 605, -340, 103, -16, 1, -6, 3, 288, -1056, 1562, -1209, 532, -134, 18, -1, -7, 20, 363, -1848, 3575, -3640, 2170, -784, 169, -20, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Row sums: A117373(n-1).
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FORMULA
| T(n,k) = 2*T(n-1,k)-T(n-2,k)-T(n-1,k-1). - R. J. Mathar, Jan 12 2011
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EXAMPLE
| 1;
2, -1;
1, -4, 1;
0, -8, 6, -1;
-1, -12, 19, -8, 1;
-2, -15, 44, -34, 10, -1;
-3, -16, 84, -104, 53, -12, 1;
-4, -14, 140, -258, 200, -76, 14, -1;
-5, -8, 210, -552,605, -340, 103, -16, 1;
-6, 3, 288, -1056, 1562, -1209, 532, -134, 18, -1;
-7, 20, 363, -1848, 3575, -3640, 2170, -784, 169, -20, 1;
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MAPLE
| A136674aux := proc(n) option remember; if n = 0 then 1; elif n= 1 then 2-x ; elif n= 2 then 1-4*x+x^2 ; else (2-x)*procname(n-1)-procname(n-2) ; end if; end proc:
A136674 := proc(n, k) coeftayl(A136674aux(n), x=0, k) ; end proc: # R. J. Mathar, Jan 12 2011
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MATHEMATICA
| (* tridiagonal matrix code*) T[n_, m_, d_] := If[ n == m, 2, If[n == d && m == d - 1, -3, If[(n == m - 1 || n == m + 1), -1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] a0 = Table[M[d], {d, 1, 10}]; Table[Det[M[d]], {d, 1, 10}]; g = Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] MatrixForm[a]; (* polynomial recursion: three initial terms necessary*) Clear[p] p[x, 0] = 1; p[x, 1] = (2 - x); p[x, 2] = 1 - 4 x + x^2; p[x_, n_] := p[x, n] = (2 - x)*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, Length[g] - 1}]
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CROSSREFS
| Sequence in context: A010247 A087605 A106246 * A144383 A205553 A178411
Adjacent sequences: A136671 A136672 A136673 * A136675 A136676 A136677
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KEYWORD
| easy,tabl,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 05 2008
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