A145661 - OEIS

A145661

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

%I #26 Jan 05 2025 19:51:39

%S 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 -111,129,-63,1,1,-13,71,-209,351,-321,127,-1,1,-15,97,-351,769,-1023,

%U 769,-255,1,1,-17,127,-545,1471,-2561,2815,-1793,511,-1,1,-19,161,-799,2561

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

%C Row sums are (-1)^(n+1)*(n-1) for n >= 1.

%C A145661, A119258 and A118801 are all essentially the same (see the Shattuck and Waldhauser paper). - Tamas Waldhauser, Jul 25 2011

%H 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 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 Triangle begins

%e 1;

%e 1, -1;

%e 1, -3, 1;

%e 1, -5, 7, -1;

%e 1, -7, 17, -15, 1;

%e 1, -9, 31, -49, 31, -1;

%e 1, -11, 49, -111, 129, -63, 1;

%e 1, -13, 71, -209, 351, -321, 127, -1;

%e 1, -15, 97, -351, 769, -1023, 769, -255, 1;

%e 1, -17, 127, -545, 1471, -2561, 2815, -1793, 511, -1;

%e 1, -19, 161, -799, 2561, -5503, 7937, -7423, 4097, -1023, 1;

%p A119258 := proc(n,k)

%p if k=0 or k = n then

%p 1;

%p elif k<0 or k> n then

%p 0;

%p else

%p 2*procname(n-1,k-1)+procname(n-1,k) ;

%p end if;

%p end proc:

%p seq(seq(A119258(n,k),k=0..n),n=0..10) ;

%p A145661 := proc(n,k)

%p (-1)^k*A119258(n,k) ;

%p end proc: # _R. J. Mathar_, Oct 21 2011

%t Clear[M, T, d, a, x, a0];

%t T[n_, m_, d_] := If[ m == n + 1, 1, If[n == d, 1, 0]];

%t M[d_] := MatrixPower[Table[T[n, m, d], {n, 1, d}, {m, 1, d}], d];

%t Table[M[d], {d, 1, 10}];

%t Table[Det[M[d]], {d, 1, 10}];

%t Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];

%t a = Join[{{1}}, Table[CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x], {n, 1, 10}]];

%t Flatten[a]

%t Join[{1}, Table[Apply[ Plus, CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}]];

%Y Cf. A135233, A112857.

%Y A193844 is an essentially identical triangle.

%K tabl,easy,sign

%O 0,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Mar 16 2009