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A189995 - OEIS
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A189995
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.
5
28, 992, 8128, 261632, 1448424448, 67100672, 1941802827776, 753623571759104, 23998307331473408, 341653284209033216, 8316321134799694594048, 740764429532373450752, 30559446583872811817762816, 496669433444154134078771167232, 17776484020396435145889494859776, 11188223110510348416175908585472
OFFSET
2,1
COMMENTS
For a(n), Milnor 2011 Theorem 5 gives the formula
2^(2*n-2)*(2^(2*n-1)-1)*numerator(4*bernoulli(n)/n)
where bernoulli(n) = abs(Bernoulli(2*n)).
See A001676 for additional comments, references, and links.
REFERENCES
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.
LINKS
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809 (see Theorem 5 and Table 3).
John W. Milnor, Spheres, Abel Prize lecture (video), 2011.
FORMULA
a(n) = 2^(2*n - 2) * (2^(2*n - 1) - 1) * abs(numerator(4*Bernoulli(2*n)/n)).
a(n) = A187595(4*n-1) for n > 1.
EXAMPLE
a(2) = 2^2 * (2^3 - 1) * abs(numerator(4 * Bernoulli(4)/2)) = 4 * 7 * abs(numerator(2 * (-1/30))) = 28
MATHEMATICA
Table[2^(2*n-2)*(2^(2*n-1)-1)*Abs[Numerator[4*BernoulliB[2*n]/n]], {n, 2, 17}]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jun 15 2011
STATUS
approved