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%I #44 Sep 08 2022 08:45:53
%S 0,1,0,-3,0,45,0,-1575,0,99225,0,-9823275,0,1404728325,0,
%T -273922023375,0,69850115960625,0,-22561587455281875,0,
%U 9002073394657468125,0,-4348001449619557104375,0,2500100833531245335015625,0,-1687568062633590601135546875
%N Expansion of e.g.f.: sin(arctan(x)).
%C Except periodic zeros for n even, and negative signs for n == 3 (mod 4), we find the same sequence as A079484 (determinant of M(2n-1) where M(k) is the k X k matrix with m(i,j)=j if i+j=k, m(i,j)=i otherwise).
%D L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.
%H G. C. Greubel, <a href="/A177698/b177698.txt">Table of n, a(n) for n = 0..450</a>
%F E.g.f.: sin(arctan(x)) = x/sqrt(1+x^2).
%F E.g.f.: x/(G(0)+x) where G(k)= 1 - 2*x/(1 + 1/G(k+1) ); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 08 2012
%F a(n) ~ 2*sin(Pi*n/2)*n^n/exp(n). - _Vaclav Kotesovec_, Sep 25 2013
%F From _Benedict W. J. Irwin_, Nov 03 2016: (Start)
%F a(n) = y(n,n), n>0, where y(m+1,n) = (n-2*m)*y(m,n), with y(0,n)=1, for all n.
%F a(n) = Real part of -i*(2*i)^n*Gamma(1 + n/2)*Gamma(n/2)/Pi. (End)
%F From _Michael Somos_, May 04 2017: (Start)
%F a(n) = -n * (n-2) * a(n-2) for all n in Z.
%F a(n) = 1 / a(-n) for all odd n in Z.
%F a(n) = n! * binomial(n-1,(n-1)/2) * 2^(1-n) * (-1)^((n-1)/2) if n is odd > 0.
%F a(2*n + 1) = (-1)^n * A079484(n). (End)
%e G.f. = x - 3*x^3 + 45*x^5 - 1575*x^7 + 99225*x^9 - 9823275*x^11 + ...
%e d^3y/dx^3 = 18/(1+x^2)^(5/2)*x^2 -3/(1+x^2)^(3/2) -15*x^4/(1+x^2)^(7/2).
%e For x = 0, we obtain a(3) = 0 - 3 + 0 = -3.
%p a:= n-> n! * coeff(series(sin(arctan(x)), x, n+1), x, n):
%p seq(a(n), n=0..30);
%t Table[n!*SeriesCoefficient[x/Sqrt[1+x^2], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Sep 25 2013 *)
%t With[{nn=30},CoefficientList[Series[Sin[ArcTan[x]],{x,0,nn}],x] Range[ 0,nn-1]!] (* _Harvey P. Dale_, Nov 30 2015 *)
%t Join[{0}, Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == (n - 2 m)*y[m], y[0] == 1}]][n], {n, 1, 20}]] (* _Benedict W. J. Irwin_, Nov 03 2016 *)
%t Join[{0}, Table[Re[-((I (2 I)^n Gamma[1 + n/2] Gamma[n/2])/Pi)], {n, 1, 20}]] (* _Benedict W. J. Irwin_, Nov 03 2016 *)
%t a[ n_] := If[ EvenQ[n], 0, I^(n - 1) n!! (n - 2)!!]; (* _Michael Somos_, May 04 2017 *)
%o (PARI) {a(n) = if( n%2==0, 0, n<0, 1 / self()(-n), n! * binomial(n-1,n\2) * 2^(1-n) * (-1)^(n\2))}; /* _Michael Somos_, May 04 2017 */
%o (PARI) x='x+O('x^50); concat([0], Vec(x/sqrt(1+x^2))) \\ _G. C. Greubel_, Sep 25 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(x/Sqrt(1+x^2))); // _G. C. Greubel_, Sep 25 2018
%Y Cf. A079484.
%K sign
%O 0,4
%A _Michel Lagneau_, May 11 2010
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