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

0,2

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

Table of n, a(n) for n=0..32.

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

Cf. A076256, A076257, A076743.

Sequence in context: A069466 A143084 A188962 * A320603 A276409 A093453

Adjacent sequences: A076738 A076739 A076740 * A076742 A076743 A076744

Keyword

sign,tabf,easy

Author

Mohammad K. Azarian, Nov 11 2002

Extensions

Edited by Dean Hickerson, Nov 28 2002

Status

approved