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%I #24 Nov 30 2022 08:16:14
%S 28,992,8128,261632,1448424448,67100672,1941802827776,753623571759104,
%T 23998307331473408,341653284209033216,8316321134799694594048,
%U 740764429532373450752,30559446583872811817762816,496669433444154134078771167232,17776484020396435145889494859776,11188223110510348416175908585472
%N The order b_{4n-1} of the cyclic group S_{4n-1}^{bp} of oriented diffeomorphism classes of smooth homotopy (4n-1)-spheres that bound parallelizable manifolds, for n > 1.
%C For a(n), Milnor 2011 Theorem 5 gives the formula
%C 2^(2*n-2)*(2^(2*n-1)-1)*numerator(4*bernoulli(n)/n)
%C where bernoulli(n) = abs(Bernoulli(2*n)).
%C See A001676 for additional comments, references, and links.
%D J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.
%H G. C. Greubel, <a href="/A189995/b189995.txt">Table of n, a(n) for n = 2..235</a>
%H John W. Milnor, <a href="http://www.ams.org/notices/201106/rtx110600804p.pdf">Differential Topology Forty-six Years Later</a>, Notices Amer. Math. Soc. 58 (2011), 804-809 (see Theorem 5 and Table 3).
%H John W. Milnor, <a href="https://www.youtube.com/watch?v=SIZd_xBiRS0">Spheres</a>, Abel Prize lecture (video), 2011.
%F a(n) = 2^(2*n - 2) * (2^(2*n - 1) - 1) * abs(numerator(4*Bernoulli(2*n)/n)).
%F a(n) = A187595(4*n-1) for n > 1.
%e a(2) = 2^2 * (2^3 - 1) * abs(numerator(4 * Bernoulli(4)/2)) = 4 * 7 * abs(numerator(2 * (-1/30))) = 28
%t Table[2^(2*n-2)*(2^(2*n-1)-1)*Abs[Numerator[4*BernoulliB[2*n]/n]],{n,2,17}]
%o (Magma) [2^(2*n-2)*(2^(2*n-1)-1)*Abs(Numerator(4*Bernoulli(2*n)/n)): n in [2..30]]; // _G. C. Greubel_, Jan 11 2018
%Y Cf. A001676, A047680, A057617, A187595, A228689.
%K nonn
%O 2,1
%A _Jonathan Sondow_, Jun 15 2011
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