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 A145661 Triangle T(n,k) = (-1)^k * A119258(n,k) read by rows, 0 <= k <= n. 5
 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; text; internal format)
 OFFSET 0,5 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 LINKS Table of n, a(n) for n=0..59. J.-F. Chamayou, A Random Difference Equation with Dufresne Variables Revisited, arXiv preprint arXiv:1410.1708 [math.PR], 2014. See Table in Section XII. M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, Fib. Quart., 48 (2010), 290-297. EXAMPLE Triangle begins 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; 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 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}]]; CROSSREFS Cf. A135233, A112857. A193844 is an essentially identical triangle. Sequence in context: A080842 A216948 A183944 * A119258 A099608 A247285 Adjacent sequences: A145658 A145659 A145660 * A145662 A145663 A145664 KEYWORD tabl,easy,sign AUTHOR Roger L. Bagula and Gary W. Adamson, Mar 16 2009 STATUS approved

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Last modified September 30 06:20 EDT 2023. Contains 365781 sequences. (Running on oeis4.)