A327883 -id:A327883 - cp's OEIS frontend

1, 1, 4, 72, 2880, 201600, 21772800, 3353011200, 697426329600, 188305108992000, 64023737057280000, 26761922089943040000, 13488008733331292160000, 8065829222532112711680000, 5646080455772478898176000000, 4573325169175707907522560000000, 4244045756995056938180935680000000

Offset: 0

Bruno Zürcher, Sep 28 2019

Even denominators of coefficients in Taylor series expansion of 2 - 2*cos(x) - 2*x*sin(x) + x^2.
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.
Numerators of the Taylor series expansion are given by A327883.
The Taylor series itself has an expansion Sum_{k>=2} (-1)^k*2*(2*k-1)/(2*k)!*x^(2*k).

			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 - ...

Cf. A052558, A171005.

Denominator[CoefficientList[ Series[2 - 2 Cos[x] - (2 x) Sin[x] + x^2, {x, 0, 33}], x][[ ;; ;; 2]]]

a(n) = {if(n<1, n==0, (2*n)!/(2*(2*n-1)))} \\ Andrew Howroyd, Oct 09 2019

a(n) = (2*n)!/(2*(2*n-1)) = n*A010050(n-1) for n >= 1.
a(n) = A171005(2*n-1) for n >= 2. - Andrew Howroyd, Oct 09 2019
a(n) = (1/2)*(2*n)!*[x^(2*n)](1 + x*arctanh(x)) for n > 0. - Peter Luschny, Oct 09 2019
D-finite with recurrence a(n) -2*n*(2*n-3)*a(n-1)=0. - R. J. Mathar, Feb 01 2022

cp's OEIS Frontend

A327882 a(n) = n(2(n-1))! for n > 0, a(0) = 1.

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