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.

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

G. C. Greubel, Table of n, a(n) for n = 2..235

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

Cf. A001676, A047680, A057617, A187595, A228689.

Sequence in context: A097579 A091549 A034904 * A228689 A218480 A162006

Adjacent sequences: A189992 A189993 A189994 * A189996 A189997 A189998

Keyword

nonn

Author

Jonathan Sondow, Jun 15 2011

Status

approved