0,2

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

The unsigned sequence 1,2,6,2,24,24,120,240,24,720,... is n-th derivative of 1/(1-x^2). 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)=n!*binomial(n+1,k); if n+k is odd, a(n,k)=0. The nonzero coefficients of the numerators starting with the highest power of x are 1; 2; 6,2; 24,24; ... In fact this is the (n-1)-st derivative of arctanh(x). - Rostislav Kollman (kollman(AT)dynasig.cz), Jan 04 2005

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.

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

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, n, 0, -1}]], #!=0&]

Cf. A076256, A076257, A076741.

Sequence in context: A096485 A125032 A131980 this_sequence A141056 A027760 A140770

Adjacent sequences: A076740 A076741 A076742 this_sequence A076744 A076745 A076746

sign,tabf,easy

Mohammad K. Azarian (azarian(AT)evansville.edu), Nov 11 2002

Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Nov 28 2002