A076257 Coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the coefficient of the highest power of x.

1, -2, 0, 6, 0, -2, -24, 0, 24, 0, 120, 0, -240, 0, 24, -720, 0, 2400, 0, -720, 0, 5040, 0, -25200, 0, 15120, 0, -720, -40320, 0, 282240, 0, -282240, 0, 40320, 0, 362880, 0, -3386880, 0, 5080320, 0, -1451520, 0, 40320, -3628800, 0, 43545600, 0, -91445760, 0, 43545600, 0

Offset: 0

Author

Mohammad K. Azarian, Nov 05 2002

Keywords

• sign
• tabl
• easy

Comments

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

Examples

			The coefficients of the numerators starting with the coefficient of the highest power of x are 1; -2,0; 6,0,-2; -24,0,24,0; ...
		

Crossrefs

Cf. A076256, A076741, A076743.

Programs

Mathematica

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

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