OFFSET
0,2
COMMENTS
Denominator of n-th derivative is (1+x^2)^(n+1), whose coefficients are the binomial coefficients, A007318.
REFERENCES
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.
LINKS
Roland Zumkeller, Formal global optimization with Taylor models, Preprint, 2006.
Roland Zumkeller, Formal global optimization with Taylor models, Thesis 2006.
FORMULA
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.
EXAMPLE
The nonzero coefficients of the numerators starting with the constant term are: 1; -2; -2,6; 24,-24; ...
MATHEMATICA
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&]
CROSSREFS
KEYWORD
sign,tabf,easy
AUTHOR
Mohammad K. Azarian, Nov 11 2002
EXTENSIONS
Edited by Dean Hickerson, Nov 28 2002
STATUS
approved