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Expansion of tan(x^2).
2

%I #21 Jan 31 2022 19:23:51

%S 2,240,483840,4704860160,140016638361600,9962283186074419200,

%T 1448674564423160954880000,386164234578290422390456320000,

%U 174195452900540085852580211589120000,125069935244132704803738682551120691200000,136166701199201243586499061842886142826905600000

%N Expansion of tan(x^2).

%H Robert Israel, <a href="/A024348/b024348.txt">Table of n, a(n) for n = 0..116</a>

%F a(n) = A000182(n)*A001813(2*n+1). - _Robert Israel_, Dec 20 2018

%F a(n) = 2 * A009767(n). - _Sean A. Irvine_, Jul 01 2019

%p S:= series(tan(x),x, 52):

%p seq(coeff(S,x,2*k+1)*(4*k+2)!,k=0..25); # _Robert Israel_, Dec 20 2018

%t With[{m=50}, CoefficientList[Series[Tan[x^2], {x,0,m}], x]*Range[0, m]!][[3;; ;; 4]] (* _G. C. Greubel_, Jan 31 2022 *)

%o (Sage) [factorial(4*n+2)*( tan(x^2) ).series(x, 4*n+3).list()[4*n+2] for n in (0..20)] # _G. C. Greubel_, Jan 31 2022

%o (Magma)

%o m:=50; R<x>:=PowerSeriesRing(Rationals(), m);

%o b:= Coefficients(R!(Laplace( Tan(x^2) )));

%o [b[4*n-3]: n in [1..Floor((m-2)/4)]]; // _G. C. Greubel_, Jan 31 2022

%Y Cf. A000182, A001813.

%K nonn

%O 0,1

%A _R. H. Hardin_

%E Extended and signs tested by _Olivier GĂ©rard_, Mar 15 1997