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A000373
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Conjectured dimension of a module associated with the free commutative Moufang loop with n generators.
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1
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0, 0, 1, 8, 44, 214, 1000, 4592, 20888, 94846, 434973, 2042836, 9979086, 51460622, 283839957, 1688139424, 10859199656, 75338888918, 560740210491, 4445766353604, 37329808482989, 330143634313064, 3064464030121369
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OFFSET
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1,4
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COMMENTS
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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
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REFERENCES
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Yu. I. Manin, Cubic Forms, Second edition, North-Holland Publishing Co., Amsterdam, 1986, page 312. MR0833513 (87d:11037)
Smith, Jonathan D. H.; Commutative Moufang loops and Bessel functions. Invent. Math. 67 (1982), no. 1, 173-187.
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LINKS
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Table of n, a(n) for n=1..23.
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EXAMPLE
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G.f. = x^3 + 8*x^4 + 44*x^5 + 214*x^6 + 1000*x^7 + 4592*x^8 + 20888*x^9 + ...
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PROG
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(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 */
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CROSSREFS
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Sequence in context: A270678 A292487 A125318 * A176688 A272154 A270935
Adjacent sequences: A000370 A000371 A000372 * A000374 A000375 A000376
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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