

A058927


Numerators of series related to triangular cacti.


2



1, 1, 5, 49, 243, 14641, 371293, 253125, 410338673, 16983563041, 1400846643, 41426511213649, 95367431640625, 617673396283947, 10260628712958602189, 756943935220796320321, 7474615974418932603, 827909024473876953125, 456487940826035155404146917, 510798409623548623605717
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OFFSET

0,3


COMMENTS

From 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 numerators were taken for this sequence and the denominators for A058928. This leads to the following
Conjecture: S(x) = Sum_{n>=0} ((2*n+1)^(n1)/(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)^(n1)), then a(n)=A052750(n)/d_n. (End)
The above conjecture is correct and follows from formula given in A034940 for the number of rooted labeled triangular cacti with 2n+1 nodes.  Andrew Howroyd, Aug 30 2018


REFERENCES

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and TreeLike Structures, Camb. 1998, p. 307.


LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200


FORMULA

G.f.: A(x) satisfies A(x)=exp(x*A(x)^2).  Vladimir Kruchinin, Feb 09 2013
a(n) = numerator(A034940(n)/(2*n+1)!) = numerator((2*n+1)^(n1)/(2^n*n!)).  Andrew Howroyd, Aug 30 2018


PROG

(PARI) a(n)={numerator((2*n+1)^(n1)/(2^n*n!))} \\ Andrew Howroyd, Aug 30 2018


CROSSREFS

Cf. A000165, A034940, A052750, A058928.
Sequence in context: A007406 A196326 A273385 * A083224 A201358 A242035
Adjacent sequences: A058924 A058925 A058926 * A058928 A058929 A058930


KEYWORD

nonn,frac,easy


AUTHOR

N. J. A. Sloane, Jan 12 2001


EXTENSIONS

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
Terms a(12) and beyond from Andrew Howroyd, Aug 30 2018


STATUS

approved



