Bruno Zürcher - cp's OEIS frontend

User: Bruno Zürcher

Bruno Zürcher has authored 2 sequences.

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

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

A290159 Numerators of coefficients in Taylor series expansion of (1+x+x^2)^(1/2).

Original entry on oeis.org

1, 1, 3, -3, 3, 15, -57, 21, 867, -1893, 1581, 8283, -76953, 34203, 361551, -869691, 6420387, 34130067, -167946159, 79445631, 1696170093, -4239570255, 4083041217, 21859150803, -442212416121, 215805655695, 2316081934929, -5909439428697, 11656013746863, 62663656767603, -322045194694305, 160129270032933, 27589357112530467

Offset: 0

Bruno Zürcher, Jul 22 2017

Denominators of the Taylor series expansion are given by A046161.
The terms after the second are divisible by 3.
The sequence of the absolute values is not monotonic.

Cf. A046161 (denominators).

a:= n-> numer(coeff(series(sqrt(1+x+x^2), x, n+3), x, n)):
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 25 2017

Numerator[CoefficientList[Series[Sqrt[1+x+x^2], {x, 0, 32}], x]]

x = 'x + O('x^40); apply(x->numerator(x), Vec((1+x+x^2)^(1/2))) \\ Michel Marcus, Jul 24 2017

cp's OEIS Frontend

User: Bruno Zürcher

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

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A290159 Numerators of coefficients in Taylor series expansion of (1+x+x^2)^(1/2).

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cp's OEIS Frontend

User: Bruno Zürcher

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

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Programs

Mathematica

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A290159 Numerators of coefficients in Taylor series expansion of (1+x+x^2)^(1/2).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

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