A076741 - OEIS

%I #9 Jun 24 2014 01:08:32

%S 1,-2,-2,6,24,-24,24,-240,120,-720,2400,-720,-720,15120,-25200,5040,

%T 40320,-282240,282240,-40320,40320,-1451520,5080320,-3386880,362880,

%U -3628800,43545600,-91445760,43545600,-3628800,-3628800,199584000,-1197504000

%N Nonzero coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the constant term.

%C Denominator of n-th derivative is (1+x^2)^(n+1), whose coefficients are the binomial coefficients, A007318.

%D Roland Zumkeller, Formal global optimization with Taylor models, IJCAR (Ulrich Furbach and Natara jan Shankar, eds.), Lecture Notes in Computer Science, vol. 4130, Springer, 2006, pp. 408-422.

%H Roland Zumkeller, <a href="http://www.lix.polytechnique.fr/~zumkeller/Publications_files/rz_formal_taylor.pdf">Formal global optimization with Taylor models</a>, Preprint, 2006.

%H Roland Zumkeller, <a href="http://roland.zumkeller.googlepages.com/thesis.html">Formal global optimization with Taylor models</a>, Thesis 2006.

%F For 0<=k<=n, let a(n, k) be the coefficient of x^k in the numerator of the n-th derivative of 1/(1+x^2). If n+k is even, a(n, k) = (-1)^((n+k)/2)*n!*binomial(n+1, k); if n+k is odd, a(n, k)=0.

%e The nonzero coefficients of the numerators starting with the constant term are: 1; -2; -2,6; 24,-24; ...

%t a[n_, k_] := Coefficient[Expand[Together[(1+x^2)^(n+1)*D[1/(1+x^2), {x, n}]]], x, k]; Select[Flatten[Table[a[n, k], {n, 0, 10}, {k, 0, n}]], #!=0&]

%Y Cf. A076256, A076257, A076743.

%K sign,tabf,easy

%O 0,2

%A _Mohammad K. Azarian_, Nov 11 2002

%E Edited by _Dean Hickerson_, Nov 28 2002