A350154 a(n) = denominator(k^n * [x^(2*n+1)] sqrt(k)*arccos(exp(-x^2/(2*k)))) for n >= 0 and fixed k > 0.

1, 12, 480, 2688, 92160, 4055040, 805109760, 148635648, 2021444812800, 9037047398400, 41855798476800, 85571854663680, 1218840851644416000, 131634811977596928000, 30539276378802487296000, 26116346696355230515200, 72745993870031978496000, 8332722934203662991360000

Offset: 0

Author

Robert B Fowler, Dec 16 2021

Keywords

• frac
• nonn

Comments

Denominators of a power series characterizing how powers of the cosine function converge to the Gaussian function.

As the cosine function is raised to increasing powers k, it converges to the Gaussian normal function. Let x be the standard deviation argument of the Gaussian function, and define a suitably scaled cosine function.

G(x) = exp(-x^2/2), Gaussian function.

C(x,k) = (cos(x/sqrt(k)))^k, k-th power of cosine function

C(x,k) - G(x) = -x^4/(12k) + x^6/(24k) - x^6/(45x^2) + ...

The usefulness of this approximation lies within the "principal half-period" of C(x,k), defined as h_k = {x : abs(x) < sqrt(k)*Pi/2}. Within h_k, k can be any real number and C(x,k) is a good approximation to G(x) even for small k, although convergence to G(x) is only reciprocal in k. Outside h_k, negative cosine values occur and the approximation deteriorates.

If we define x(k) such that G(x) = C(x(k),k) then

x = lim_{k->infinity} x(k).

The value of x(k) can be expressed as a polynomial in integer powers of x and k and coefficients A350194(n)/a(n), and characterizes how closely cosine powers approximate and converge to the Gaussian function.

Examples

			x(k) = x - (1/12)*(x^3/k) + (1/480)*(x^5/k^2) + (1/2688)*(x^7/k^3) - (1/92160)*(x^9/k^4) - (19/4055040)*(x^11/k^5) + (79/805109760)*(x^13/k^6) ...
		

Links

• David Broadhurst, Relations between A241885/A242225, A222411/A222412, and A350194/A350154.

Crossrefs

Cf. A350194, A222411/A222412, A241885/A242225.

Programs

Maple

gf := sqrt(k)*arccos(exp(-x^2/(2*k))): assume(k > 0): assume(x > 0):
ser := series(gf, x, 80): seq(denom(k^n*coeff(ser, x, 2*n+1)), n=0..17); # Peter Luschny, Dec 19 2021

Formula

The definitions of G(x) and C(x,k) lead directly to the equation

x(k) = sqrt(k)*arccos(exp(-x^2/(2k))),

which can be expanded into the power series

x(k) = Sum_{n>=0} (x^(2n+1)/k^n) * (A350194(n)/a(n)).

Theorem: A241885(n)/A242225(n) = n!*A222411(n)/(A222412(n)*(-1)^n/(1-2*n)) = n!*A350194(n)/(A350154(n)*(2*n+1)). - David Broadhurst, Apr 23 2022 (see Link).