A076741 Nonzero coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the constant term.

1, -2, -2, 6, 24, -24, 24, -240, 120, -720, 2400, -720, -720, 15120, -25200, 5040, 40320, -282240, 282240, -40320, 40320, -1451520, 5080320, -3386880, 362880, -3628800, 43545600, -91445760, 43545600, -3628800, -3628800, 199584000, -1197504000

Offset: 0

Author

Mohammad K. Azarian, Nov 11 2002

Keywords

• sign
• tabf
• easy

Comments

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

Examples

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

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.

Crossrefs

Cf. A076256, A076257, A076743.

Programs

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&]

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.

Extensions

Edited by Dean Hickerson, Nov 28 2002