This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372324 #20 May 02 2024 06:11:32
%S A372324 0,0,1,3,16,80,544,3808,32768,294912,3096576,34062336,423493632,
%T A372324 5505417216,79199207424,1187988111360,19423989596160,330207823134720,
%U A372324 6050282848911360,114955374129315840,2333627101111910400,49006169123350118400,1091943568123940044800
%N A372324 Expansion of e.g.f. arcsin(x)^2/(2*(1 - x)).
%C A372324 a(2*n) appears in the formula for the limit, as k -> infinity, of the area between cos^(2*n)(x) and cos^(2*n)(k*x) on the interval [0, Pi]. To be precise, here is the formula: a(2*n)*(16/Pi)/((2*n)!!)^2 = Lim_{k->oo} Integral_{x=0..Pi} abs(cos^(2*n)(x) - cos^(2*n)(k*x)) dx. See the article by Dombrowski and Dresden.
%H A372324 Muhammad Adam Dombrowski and Gregory Dresden, <a href="https://arxiv.org/abs/2404.17694">Areas Between Cosines</a>, arXiv:2404.17694 [math.CO], 2024.
%F A372324 a(2*n+1) = (2*n+1)*a(2*n).
%F A372324 a(2*n) = (2*n)*(2*n-1)*a(2*n-2) + ((2*n-2)!!)^2.
%F A372324 a(n) = (n!)*Sum_{k=0..(n-2)/2} ((2*k)!!)/(((2*k+1)!!)*(2*k+2)).
%F A372324 E.g.f.: arcsin(x)^2/(2*(1 - x)).
%F A372324 a(n) ~ n! * (Pi^2/8) * (1 - 2^(5/2)/(Pi^(3/2)*sqrt(n))). - _Vaclav Kotesovec_, May 01 2024
%F A372324 D-finite with recurrence a(n) -n*a(n-1) -(n-2)^2*a(n-2) +(n-2)^3*a(n-3)=0. - _R. J. Mathar_, May 02 2024
%t A372324 Table[n! SeriesCoefficient[ArcSin[x]^2/(2 (1 - x)), {x, 0, n}], {n, 0, 22}]
%Y A372324 Cf. A296726.
%K A372324 nonn
%O A372324 0,4
%A A372324 _Greg Dresden_, Apr 27 2024