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 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; text; 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 LINKS 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: 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

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Last modified April 5 23:14 EDT 2020. Contains 333260 sequences. (Running on oeis4.)