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A145661 Triangle T(n,k) = (-1)^k * A119258(n,k) read by rows, 0 <= k <= n.

%I A145661 #26 Jan 05 2025 19:51:39
%S A145661 1,1,-1,1,-3,1,1,-5,7,-1,1,-7,17,-15,1,1,-9,31,-49,31,-1,1,-11,49,
%T A145661 -111,129,-63,1,1,-13,71,-209,351,-321,127,-1,1,-15,97,-351,769,-1023,
%U A145661 769,-255,1,1,-17,127,-545,1471,-2561,2815,-1793,511,-1,1,-19,161,-799,2561
%N A145661 Triangle T(n,k) = (-1)^k * A119258(n,k) read by rows, 0 <= k <= n.
%C A145661 Row sums are (-1)^(n+1)*(n-1) for n >= 1.
%C A145661 A145661, A119258 and A118801 are all essentially the same (see the Shattuck and Waldhauser paper). - Tamas Waldhauser, Jul 25 2011
%H A145661 J.-F. Chamayou, <a href="http://arxiv.org/abs/1410.1708">A Random Difference Equation with Dufresne Variables Revisited</a>, arXiv preprint arXiv:1410.1708 [math.PR], 2014. See Table in Section XII.
%H A145661 M. Shattuck and T. Waldhauser, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/48-4/Shattuck_Waldhauser.pdf">Proofs of some binomial identities using the method of last squares</a>, Fib. Quart., 48 (2010), 290-297.
%e A145661 Triangle begins
%e A145661   1;
%e A145661   1,  -1;
%e A145661   1,  -3,   1;
%e A145661   1,  -5,   7,   -1;
%e A145661   1,  -7,  17,  -15,    1;
%e A145661   1,  -9,  31,  -49,   31,    -1;
%e A145661   1, -11,  49, -111,  129,   -63,    1;
%e A145661   1, -13,  71, -209,  351,  -321,  127,    -1;
%e A145661   1, -15,  97, -351,  769, -1023,  769,  -255,    1;
%e A145661   1, -17, 127, -545, 1471, -2561, 2815, -1793,  511,    -1;
%e A145661   1, -19, 161, -799, 2561, -5503, 7937, -7423, 4097, -1023, 1;
%p A145661 A119258 := proc(n,k)
%p A145661         if k=0 or k = n then
%p A145661                 1;
%p A145661         elif k<0 or k> n then
%p A145661                 0;
%p A145661         else
%p A145661                 2*procname(n-1,k-1)+procname(n-1,k) ;
%p A145661         end if;
%p A145661 end proc:
%p A145661 seq(seq(A119258(n,k),k=0..n),n=0..10) ;
%p A145661 A145661 := proc(n,k)
%p A145661         (-1)^k*A119258(n,k) ;
%p A145661 end proc: # _R. J. Mathar_, Oct 21 2011
%t A145661 Clear[M, T, d, a, x, a0];
%t A145661 T[n_, m_, d_] := If[ m == n + 1, 1, If[n == d, 1, 0]];
%t A145661 M[d_] := MatrixPower[Table[T[n, m, d], {n, 1, d}, {m, 1, d}], d];
%t A145661 Table[M[d], {d, 1, 10}];
%t A145661 Table[Det[M[d]], {d, 1, 10}];
%t A145661 Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
%t A145661 a = Join[{{1}}, Table[CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x], {n, 1, 10}]];
%t A145661 Flatten[a]
%t A145661 Join[{1}, Table[Apply[ Plus, CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}]];
%Y A145661 Cf. A135233, A112857.
%Y A145661 A193844 is an essentially identical triangle.
%K A145661 tabl,easy,sign
%O A145661 0,5
%A A145661 _Roger L. Bagula_ and _Gary W. Adamson_, Mar 16 2009