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COMMENTS
| Contribution by L. Edson Jeffery, Jan 09 2012 (Start):
The reference [Bergeron, et al.] lists the first few terms of the relevant series as S(x)=x+(1/2)*x^3+(5/8)*x^5+(49/48)*x^7+(243/128)*x^9+..., from which the denominators were taken for this sequence and the numerators for A058927. This leads to the following
Conjecture: S(x)=sum[n=0,1,..., ((2*n+1)^(n-1)/(n!*2^n))*x^(2*n+1)] = (A052750(n)/A000165(n))*x^(2*n+1). Letting D_n be the set of divisors of n! and d_n=max(k in D_n : k | (2*n+1)^(n-1)), then a(n)=A000165(n)/d_n, with the next few terms, for n=12,...,20, being:
{78479622144, 209924915200, 1428329123020800, 42849873690624000, 170993385472000, 7611536747003904, 1678343852714360832000, 747740921331712000, 2551082656125828464640000}. (End)
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REFERENCES
| F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 307.
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