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%I #24 Feb 01 2022 13:41:33
%S 1,1,4,72,2880,201600,21772800,3353011200,697426329600,
%T 188305108992000,64023737057280000,26761922089943040000,
%U 13488008733331292160000,8065829222532112711680000,5646080455772478898176000000,4573325169175707907522560000000,4244045756995056938180935680000000
%N a(n) = n*(2*(n-1))! for n > 0, a(0) = 1.
%C Even denominators of coefficients in Taylor series expansion of 2 - 2*cos(x) - 2*x*sin(x) + x^2.
%C Equivalent to the even denominators of expansion of (1 - cos(x)^2 + (x-sin(x))^2, which is the square of the secant length measured from the origin (0,0) to the cycloid point (1-cos(x), x-sin(x)). Note that only x^4 has the first nonzero coefficient of the series.
%C Numerators of the Taylor series expansion are given by A327883.
%C The Taylor series itself has an expansion Sum_{k>=2} (-1)^k*2*(2*k-1)/(2*k)!*x^(2*k).
%F a(n) = (2*n)!/(2*(2*n-1)) = n*A010050(n-1) for n >= 1.
%F a(n) = A171005(2*n-1) for n >= 2. - _Andrew Howroyd_, Oct 09 2019
%F a(n) = (1/2)*(2*n)!*[x^(2*n)](1 + x*arctanh(x)) for n > 0. - _Peter Luschny_, Oct 09 2019
%F D-finite with recurrence a(n) -2*n*(2*n-3)*a(n-1)=0. - _R. J. Mathar_, Feb 01 2022
%e 2 + x^2 - 2*cos(x) - 2*x*sin(x) = (1/4)*x^4 - (1/72)*x^6 + (1/2880)*x^8 - (1/201600)*x^10 + (1/21772800)*x^12 - ...
%t Denominator[CoefficientList[ Series[2 - 2 Cos[x] - (2 x) Sin[x] + x^2, {x, 0, 33}], x][[ ;; ;; 2]]]
%o (PARI) a(n) = {if(n<1, n==0, (2*n)!/(2*(2*n-1)))} \\ _Andrew Howroyd_, Oct 09 2019
%Y Cf. A052558, A171005.
%K nonn,easy
%O 0,3
%A _Bruno Zürcher_, Sep 28 2019
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