

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
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OFFSET

0,2


COMMENTS

Denominator of nth 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 nth derivative of 1/(1x^2). For 0<=k<=n, let a(n,k) be the coefficient of x^k in the numerator of the nth derivative of 1/(1x^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 (n1)st derivative of arctanh(x).  Rostislav Kollman (kollman(AT)dynasig.cz), Jan 04 2005


LINKS

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


FORMULA

For 0<=k<=n, let a(n, k) be the coefficient of x^k in the numerator of the nth 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: A320016 A096485 A125032 * A131980 A217448 A280705
Adjacent sequences: A076740 A076741 A076742 * A076744 A076745 A076746


KEYWORD

sign,tabf,easy


AUTHOR

Mohammad K. Azarian, Nov 11 2002


EXTENSIONS

Edited by Dean Hickerson, Nov 28 2002


STATUS

approved



