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A145661
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Triangle T(n,k) = (-1)^k * A119258(n,k) read by rows, 0<=k <=n.
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3
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1, 1, -1, 1, -3, 1, 1, -5, 7, -1, 1, -7, 17, -15, 1, 1, -9, 31, -49, 31, -1, 1, -11, 49, -111, 129, -63, 1, 1, -13, 71, -209, 351, -321, 127, -1, 1, -15, 97, -351, 769, -1023, 769, -255, 1, 1, -17, 127, -545, 1471, -2561, 2815, -1793, 511, -1, 1, -19, 161, -799, 2561
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are (-1)^(n+1)*(n-1) for n>=1.
A145661, A119258 and A118801 are all essentially the same (see the Shattuck and Waldhauser paper). - Tamas Waldhauser, Jul 25 2011
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LINKS
| M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, Fib. Quart., 48 (2010), 290-297.
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EXAMPLE
| 1;
1, -1;
1, -3, 1;
1, -5, 7, -1;
1, -7, 17, -15, 1;
1, -9, 31, -49, 31, -1;
1, -11, 49, -111, 129, -63, 1;
1, -13, 71, -209, 351, -321, 127, -1;
1, -15, 97, -351, 769, -1023, 769, -255, 1;
1, -17, 127, -545, 1471, -2561, 2815, -1793, 511, -1;
1, -19, 161, -799, 2561, -5503, 7937, -7423, 4097, -1023, 1;
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MAPLE
| A119258 := proc(n, k)
if k=0 or k = n then
1;
elif k<0 or k> n then
0;
else
2*procname(n-1, k-1)+procname(n-1, k) ;
end if;
end proc:
seq(seq(A119258(n, k), k=0..n), n=0..10) ;
A145661 := proc(n, k)
(-1)^k*A119258(n, k) ;
end proc: # R. J. Mathar, Oct 21 2011
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MATHEMATICA
| Clear[M, T, d, a, x, a0];
T[n_, m_, d_] := If[ m == n + 1, 1, If[n == d, 1, 0]];
M[d_] := MatrixPower[Table[T[n, m, d], {n, 1, d}, {m, 1, d}], d];
Table[M[d], {d, 1, 10}];
Table[Det[M[d]], {d, 1, 10}];
Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
a = Join[{{1}}, Table[CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x], {n, 1, 10}]];
Flatten[a]
Join[{1}, Table[Apply[ Plus, CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}]];
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CROSSREFS
| Cf. A135233, A112857.
Sequence in context: A204027 A080842 A183944 * A119258 A099608 A047969
Adjacent sequences: A145658 A145659 A145660 * A145662 A145663 A145664
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KEYWORD
| tabl,easy,sign
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Mar 16 2009.
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