|
| |
|
|
A076743
|
|
Nonzero coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the highest power of x.
|
|
4
| |
|
|
1, -2, 6, -2, -24, 24, 120, -240, 24, -720, 2400, -720, 5040, -25200, 15120, -720, -40320, 282240, -282240, 40320, 362880, -3386880, 5080320, -1451520, 40320, -3628800, 43545600, -91445760, 43545600, -3628800, 39916800, -598752000, 1676505600
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| 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
|
|
|
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 highest power of x are: 1; -2; 6,-2; -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, n, 0, -1}]], #!=0&]
|
|
|
CROSSREFS
| Cf. A076256, A076257, A076741.
Sequence in context: A096485 A125032 A131980 * A027760 A141056 A141498
Adjacent sequences: A076740 A076741 A076742 * A076744 A076745 A076746
|
|
|
KEYWORD
| sign,tabf,easy
|
|
|
AUTHOR
| Mohammad K. Azarian (azarian(AT)evansville.edu), Nov 11 2002
|
|
|
EXTENSIONS
| Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Nov 28 2002
|
| |
|
|
