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A076743
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Nonzero coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the highest power of x.
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4
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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
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0,2
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COMMENTS
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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
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LINKS
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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 nonzero coefficients of the numerators starting with the highest power of x are: 1; -2; 6,-2; -24,24; ...
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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]; Select[Flatten[Table[a[n, k], {n, 0, 10}, {k, n, 0, -1}]], #!=0&]
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CROSSREFS
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KEYWORD
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sign,tabf,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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