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, 2, 4, 7, 13, 24, 44, 81, 151, 280, 525, 984, 1859, 3511, 6682, 12709, 24334, 46565, 89626, 172381, 333262, 643733, 1249147, 2421592, 4713715, 9165792, 17888456, 34873456, 68212220, 133269997, 261167821, 511211652, 1003436520, 1967293902

Offset

1,2

Comments

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

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).

Links

Matthew House, Table of n, a(n) for n = 1..3328

Sergi Elizalde, Improved bounds on the number of numerical semigroups of a given genus, arXiv:0905.0489 [math.CO], May 4, 2009. See Table 1, p. 8.

Prog

(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)$
makelist(ratcoef(gf, x, n), n, 1, 100); /* Franck Maminirina Ramaharo, Jan 15 2019 */

Crossrefs

Cf. A000045, A007323.

Sequence in context: A305442 A000073 A255069 * A276661 A005318 A102111

Adjacent sequences: A160251 A160252 A160253 * A160255 A160256 A160257

Keyword

nonn,easy

Author

Jonathan Vos Post, May 06 2009

Extensions

Edited, and name replaced by the g.f. by Franck Maminirina Ramaharo, Jan 15 2019

Status

approved