A006298 Number of genus 2 rooted maps with 1 face with n vertices.

21, 483, 6468, 66066, 570570, 4390386, 31039008, 205633428, 1293938646, 7808250450, 45510945480, 257611421340, 1422156202740, 7683009544980, 40729207226400, 212347275857640, 1090848505817070, 5530195966465170, 27704671055301240, 137308238124957900, 673903972248687180

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Call C(p,[alpha],g) the number of partitions of the cyclically ordered set [p], of cyclic type [alpha], and of genus g (genus g Faa di Bruno coefficients of type [alpha]). The number C(2n,[2^n],g) of genus g partitions of the set [2n] into n blocks of length 2 is given by the coefficient of u^(2g) in the power series expansion of ((2*k)!/((k+1)!*(k-2g)!))*((u/2)/tanh(u/2))^(k+1) about the point u=0 [Harer-Zagier]. The given sequence a(n) is C(2n,[2^n],2), or, equivalently, it is the number of genus 2 partitions of the set [2n] into n parts with no singletons; it vanishes for n < 4 and a(4)=21. - Robert Coquereaux, Mar 07 2024

Examples

			G.f. = 21*x^4 + 483*x^5 + 6468*x^6 + 66066*x^7 + 570570*x^8 + 4390386*x^9 + ...
		

References

• J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986) 475-485.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Links

• G. C. Greubel, Table of n, a(n) for n = 4..1000
• Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus: A compendium of results, Journal of Integer Sequences, Vol. 27 (2024), article 24.2.6. See p.19.
• Robert Cori and G Hetyei, Counting partitions of a fixed genus, arXiv preprint arXiv:1710.09992 [math.CO], 2017.
• T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
• Liang Zhao and Fengyao Yan, Note on Total Positivity for a Class of Recursive Matrices, Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.5.
• Jean-Bernard Zuber, Counting partitions by genus. I. Genus 0 to 2, Enumer. Comb. Appl. 4 (2) (2024), article #S2R13. See pp. 14-17.

Crossrefs

Cf. A035309.

Cf. A000108 for C(2n, [2^n], 0) and A002802 for C(2n, [2^n], 1).

Programs

Maple

gf := 21*x^4*(x + 1)*(1 - 4*x)^(-11/2): ser := series(gf, x, 32):
seq(coeff(ser, x, n), n = 4..24);  # Peter Luschny, Mar 07 2024

Mathematica

CoefficientList[Series[21*x^4*(1 + x)/Sqrt[(1 - 4*x)^11], {x, 0, 50}]/x^4, x] (* G. C. Greubel, Jan 30 2017 *)
a[n_] := ((-2 + 5 * n) * (2 * n)!)/(1440 * n! * (n - 4)!) (* Robert Coquereaux, Mar 07 2024 *)

PARI

A006298(n) = if(n<4,0,if(n==4,21,((5*(n-1)+3)*(4*(n-1)+2)*A006298(n-1))/((5*(n-1)-2)*((n-1)-3)))); \\ Joerg Arndt, Apr 07 2013

PARI

x='x+O('x^66);  Vec(21*x^4*(1+x)/sqrt((1-4*x)^11)) \\ Joerg Arndt, Apr 07 2013

Formula

D-finite with recurrence a(n+1) = ((5*n+3)*(4n*+2)*a(n))/((5*n-2)(n-3)).

G.f.: 21*x^4*(1+x)/sqrt((1-4*x)^11). a(n) = 21 * (A020922(n-4) + A020922(n-3)). - Ralf Stephan, Mar 13 2004 (g.f. corrected by Joerg Arndt, Apr 07 2013)

0 = a(n)*(+16*a(n+1) +62*a(n+2) +6*a(n+3)) +a(n+1)*(-38*a(n+1) -5*a(n+2) +17*a(n+3)) +a(n+2)*(-23*a(n+2) +a(n+3)) for all n in Z. - Michael Somos, Mar 30 2016

a(n) ~ n^(9/2) * 2^(2*n-5) / (9*sqrt(Pi)). - Vaclav Kotesovec, Mar 30 2016

a(n) = ((-2+5*n)*(2*n)!)/(1440*n!*(n-4)!) for n >= 4. - Robert Coquereaux, Mar 07 2024