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%I #13 May 22 2014 22:26:02
%S 0,0,1,8,44,214,1000,4592,20888,94846,434973,2042836,9979086,51460622,
%T 283839957,1688139424,10859199656,75338888918,560740210491,
%U 4445766353604,37329808482989,330143634313064,3064464030121369
%N Conjectured dimension of a module associated with the free commutative Moufang loop with n generators.
%C 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
%D Yu. I. Manin, Cubic Forms, Second edition, North-Holland Publishing Co., Amsterdam, 1986, page 312. MR0833513 (87d:11037)
%D Smith, Jonathan D. H.; Commutative Moufang loops and Bessel functions. Invent. Math. 67 (1982), no. 1, 173-187.
%e G.f. = x^3 + 8*x^4 + 44*x^5 + 214*x^6 + 1000*x^7 + 4592*x^8 + 20888*x^9 + ...
%o (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 */
%K nonn
%O 1,4
%A _N. J. A. Sloane_
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