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A160254 Expansion of x*(2 - 3*x + x^2 - 4*x^3 + 3*x^4 - 2*x^5 + x*(1 - x - x^3)*sqrt((1 + 2*x)/(1 - 2*x)))/(2*(1 - 3*x + 3*x^2 - 3*x^3 + 4*x^4 - 3*x^5 + 2*x^6)). 1

%I

%S 1,2,4,7,13,24,44,81,151,280,525,984,1859,3511,6682,12709,24334,46565,

%T 89626,172381,333262,643733,1249147,2421592,4713715,9165792,17888456,

%U 34873456,68212220,133269997,261167821,511211652,1003436520,1967293902

%N Expansion of x*(2 - 3*x + x^2 - 4*x^3 + 3*x^4 - 2*x^5 + x*(1 - x - x^3)*sqrt((1 + 2*x)/(1 - 2*x)))/(2*(1 - 3*x + 3*x^2 - 3*x^3 + 4*x^4 - 3*x^5 + 2*x^6)).

%C a(n) is the number of nodes at level n in certain generating tree, denoted C, that embeds the tree of numerical semigroups.

%C Elizalde (2009) established that the number A007323(n) of numerical semigroups of genus n is bounded in C as follows: A000045(n+2) - 1 <= A007323(n) <= a(n) <= 1 + 3*2^(n - 3).

%H Matthew House, <a href="/A160254/b160254.txt">Table of n, a(n) for n = 1..3328</a>

%H Sergi Elizalde, <a href="https://arxiv.org/abs/0905.0489">Improved bounds on the number of numerical semigroups of a given genus</a>, arXiv:0905.0489 [math.CO], May 4, 2009. See Table 1, p. 8.

%o (Maxima) gf : taylor(x*(2 - 3*x + x^2 - 4*x^3 + 3*x^4 - 2*x^5 + x*(1 - x - x^3)*sqrt((1 + 2*x)/(1 - 2*x)))/(2*(1 - 3*x + 3*x^2 - 3*x^3 + 4*x^4 - 3*x^5 + 2*x^6)), x, 0, 100)$

%o makelist(ratcoef(gf, x, n), n, 1, 100); /* _Franck Maminirina Ramaharo_, Jan 15 2019 */

%Y Cf. A000045, A007323.

%K nonn,easy

%O 1,2

%A _Jonathan Vos Post_, May 06 2009

%E Edited, and name replaced by the g.f. by _Franck Maminirina Ramaharo_, Jan 15 2019

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Last modified December 14 22:42 EST 2019. Contains 329987 sequences. (Running on oeis4.)