A000373 Conjectured dimension of a module associated with the free commutative Moufang loop with n generators.

0, 0, 1, 8, 44, 214, 1000, 4592, 20888, 94846, 434973, 2042836, 9979086, 51460622, 283839957, 1688139424, 10859199656, 75338888918, 560740210491, 4445766353604, 37329808482989, 330143634313064, 3064464030121369

Offset

1,4

Comments

There is an explicit formula for the n-th term of this sequence (see Eq. (8.4) of Smith (1982)). It is conjectured that this gives the answer to a question of Manin about the dimension of a certain module associated with the free commutative Moufang loop with n generators. - N. J. A. Sloane, May 21 2014

The underlying hypothesis has been disproven, see Grishkov & Shestakov (2008-2011). - Matthew House, Sep 03 2024

References

Yu. I. Manin, Cubic Forms, Second edition, North-Holland Publishing Co., Amsterdam, 1986, page 312. MR0833513 (87d:11037)

Links

Matthew House, Table of n, a(n) for n = 1..523

Alexander N. Grishkov and Ivan P. Shestakov, Commutative Moufang loops and alternative algebras, J. Algebra, 333 (2011), 1-13; Preprint, arXiv:0811.3787 [math.RA], 2008.

Jonathan D. H. Smith, Commutative Moufang loops and Bessel functions, Invent. Math. 67 (1982), no. 1, 173-187.

Example

G.f. = x^3 + 8*x^4 + 44*x^5 + 214*x^6 + 1000*x^7 + 4592*x^8 + 20888*x^9 + ...

Prog

(PARI) {a(n) = local(A); if( n<3, 0, A = Vec(-1 + serlaplace( serlaplace( subst( 1 / besselj(0, x + O(x^n)), x^2, 4*x)))); A[1] = 0; sum(k=1, (n-1)\2, sum(p=0, n - 2*k - 1, n! / p! / (2*k+1)! / (n - p - 2*k -1 )! * (A[k] + binomial( p+k-1, k-1)))))}; /* Michael Somos, May 17 2004 */

Crossrefs

Cf. A152123.

Sequence in context: A270678 A292487 A125318 * A176688 A272154 A270935

Adjacent sequences: A000370 A000371 A000372 * A000374 A000375 A000376

Keyword

nonn

Author

N. J. A. Sloane

Status

approved