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%I M5117 #75 Mar 08 2024 15:37:44
%S 21,483,6468,66066,570570,4390386,31039008,205633428,1293938646,
%T 7808250450,45510945480,257611421340,1422156202740,7683009544980,
%U 40729207226400,212347275857640,1090848505817070,5530195966465170,27704671055301240,137308238124957900,673903972248687180
%N Number of genus 2 rooted maps with 1 face with n vertices.
%C 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
%D J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986) 475-485.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.
%H G. C. Greubel, <a href="/A006298/b006298.txt">Table of n, a(n) for n = 4..1000</a>
%H Robert Coquereaux and Jean-Bernard Zuber, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Coquereaux/coque5.html">Counting partitions by genus: A compendium of results</a>, Journal of Integer Sequences, Vol. 27 (2024), article 24.2.6. See p.19.
%H Robert Cori and G Hetyei, <a href="http://arxiv.org/abs/1710.09992">Counting partitions of a fixed genus</a>, arXiv preprint arXiv:1710.09992 [math.CO], 2017.
%H T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(72)90056-1">Counting rooted maps by genus</a>, J. Comb. Thy B13 (1972), 122-141 and 192-218.
%H Liang Zhao and Fengyao Yan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Zhao/zhao17.html">Note on Total Positivity for a Class of Recursive Matrices</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.5.
%H Jean-Bernard Zuber, <a href="https://doi.org/10.54550/ECA2024V4S2R13">Counting partitions by genus. I. Genus 0 to 2</a>, Enumer. Comb. Appl. 4 (2) (2024), article #S2R13. See pp. 14-17.
%F D-finite with recurrence a(n+1) = ((5*n+3)*(4n*+2)*a(n))/((5*n-2)(n-3)).
%F 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)
%F 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
%F a(n) ~ n^(9/2) * 2^(2*n-5) / (9*sqrt(Pi)). - _Vaclav Kotesovec_, Mar 30 2016
%F a(n) = ((-2+5*n)*(2*n)!)/(1440*n!*(n-4)!) for n >= 4. - _Robert Coquereaux_, Mar 07 2024
%e G.f. = 21*x^4 + 483*x^5 + 6468*x^6 + 66066*x^7 + 570570*x^8 + 4390386*x^9 + ...
%p gf := 21*x^4*(x + 1)*(1 - 4*x)^(-11/2): ser := series(gf, x, 32):
%p seq(coeff(ser, x, n), n = 4..24); # _Peter Luschny_, Mar 07 2024
%t CoefficientList[Series[21*x^4*(1 + x)/Sqrt[(1 - 4*x)^11], {x, 0, 50}]/x^4, x] (* _G. C. Greubel_, Jan 30 2017 *)
%t a[n_] := ((-2 + 5 * n) * (2 * n)!)/(1440 * n! * (n - 4)!) (* _Robert Coquereaux_, Mar 07 2024 *)
%o (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
%o (PARI) x='x+O('x^66); Vec(21*x^4*(1+x)/sqrt((1-4*x)^11)) \\ _Joerg Arndt_, Apr 07 2013
%Y Cf. A035309.
%Y Cf. A000108 for C(2n, [2^n], 0) and A002802 for C(2n, [2^n], 1).
%K nonn,easy
%O 4,1
%A _N. J. A. Sloane_
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