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Search: id:A076257
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| A076257 |
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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. |
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+0
4
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| 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
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Denominator of n-th derivative is (1+x^2)^(n+1), whose coefficients are the binomial coefficients, A007318.
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FORMULA
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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.
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EXAMPLE
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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; ...
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MATHEMATICA
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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}]]
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CROSSREFS
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Cf. A076256, A076741, A076743.
Sequence in context: A047918 A138701 A050821 this_sequence A162974 A046520 A157195
Adjacent sequences: A076254 A076255 A076256 this_sequence A076258 A076259 A076260
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KEYWORD
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sign,tabl,easy
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AUTHOR
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Mohammad K. Azarian (azarian(AT)evansville.edu), Nov 05 2002
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EXTENSIONS
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Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Nov 28 2002
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