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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.

%I A350154 #41 Apr 24 2022 20:47:54
%S A350154 1,12,480,2688,92160,4055040,805109760,148635648,2021444812800,
%T A350154 9037047398400,41855798476800,85571854663680,1218840851644416000,
%U A350154 131634811977596928000,30539276378802487296000,26116346696355230515200,72745993870031978496000,8332722934203662991360000
%N 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.
%C A350154 Denominators of a power series characterizing how powers of the cosine function converge to the Gaussian function.
%C A350154 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.
%C A350154     G(x) = exp(-x^2/2),             Gaussian function.
%C A350154     C(x,k) = (cos(x/sqrt(k)))^k,    k-th power of cosine function
%C A350154     C(x,k) - G(x) = -x^4/(12k) + x^6/(24k) - x^6/(45x^2) + ...
%C A350154 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.
%C A350154 If we define x(k) such that G(x) = C(x(k),k) then
%C A350154     x = lim_{k->infinity} x(k).
%C A350154 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.
%H A350154 David Broadhurst, <a href="/A241885/a241885.txt">Relations between A241885/A242225, A222411/A222412, and A350194/A350154.</a>
%F A350154 The definitions of G(x) and C(x,k) lead directly to the equation
%F A350154   x(k) = sqrt(k)*arccos(exp(-x^2/(2k))),
%F A350154 which can be expanded into the power series
%F A350154   x(k) = Sum_{n>=0} (x^(2n+1)/k^n) * (A350194(n)/a(n)).
%F A350154 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).
%e A350154 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) ...
%p A350154 gf := sqrt(k)*arccos(exp(-x^2/(2*k))): assume(k > 0): assume(x > 0):
%p A350154 ser := series(gf, x, 80): seq(denom(k^n*coeff(ser, x, 2*n+1)), n=0..17); # _Peter Luschny_, Dec 19 2021
%Y A350154 Cf. A350194, A222411/A222412, A241885/A242225.
%K A350154 frac,nonn
%O A350154 0,2
%A A350154 _Robert B Fowler_, Dec 16 2021