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A006298
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Number of genus 2 rooted maps with 1 face with n vertices.
(Formerly M5117)
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13
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21, 483, 6468, 66066, 570570, 4390386, 31039008, 205633428, 1293938646, 7808250450, 45510945480, 257611421340, 1422156202740, 7683009544980, 40729207226400, 212347275857640, 1090848505817070, 5530195966465170, 27704671055301240, 137308238124957900, 673903972248687180
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OFFSET
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4,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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D-finite with recurrence a(n+1) = ((5*n+3)*(4n*+2)*a(n))/((5*n-2)(n-3)).
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) = ((-2+5*n)*(2*n)!)/(1440*n!*(n-4)!) for n >= 4. - Robert Coquereaux, Mar 07 2024
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EXAMPLE
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G.f. = 21*x^4 + 483*x^5 + 6468*x^6 + 66066*x^7 + 570570*x^8 + 4390386*x^9 + ...
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MAPLE
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gf := 21*x^4*(x + 1)*(1 - 4*x)^(-11/2): ser := series(gf, x, 32):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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