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A007137
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Number of rooted maps with n edges on the projective plane.
(Formerly M4734)
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5
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1, 10, 98, 982, 10062, 105024, 1112757, 11934910, 129307100, 1412855500, 15548498902, 172168201088, 1916619748084, 21436209373224, 240741065193282, 2713584138389838, 30687358107371442, 348061628432108352
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OFFSET
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1,2
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REFERENCES
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E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
David M. Jackson and Terry I. Visentin, An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces, Chapman & Hall/CRC, circa 2000. See page 227.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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From Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005: (Start)
G.f.: ((2*R+1)/3-sqrt(R*(R+2)/3))/(2*x) where R=sqrt(1-12*x);
a(n) ~ sqrt(3/2)*12^n/(n^(5/4)*GAMMA(3/4)). (End)
a(n) = (2/(n+1)) * Sum_{k=0..n-1} binomial(2*n, k) * 3^k * A002426(n-k).
G.f. y=A(x) satisfies:
0 = 9*x^3*y^4 - 6*x^2*y^3 + 2*x*(21*x - 1)*y^2 + (10*x - 1)*y + x.
0 = x*(4*x + 1)*(12*x - 1)^3*y'''' + 4*(132*x^2 + 19*x - 1)*(12*x - 1)^2*y''' + 12*(1476*x^2 + 60*x - 11)*(12*x - 1)*y'' + 72*(2016*x^2 - 117*x - 4)*y' + 648*(16*x - 1)*y.
(End)
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MAPLE
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R:=sqrt(1-12*x): seq(coeff(convert(series(((2*R+1)/3-sqrt(R*(R+2)/3))/(2*x), x, 50), polynom), x, n), n=1..25); # Pab Ter, Nov 07 2005
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MATHEMATICA
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With[{r=Sqrt[1-12x]}, Rest[CoefficientList[Series[((2r+1)/3-Sqrt[r (r+2)/3])/ (2x), {x, 0, 20}], x]]](* Harvey P. Dale, Mar 02 2018 *)
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PROG
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(PARI)
seq(N) = {
my(x = 'x + O('x^(N+2)), r=sqrt(1-12*x));
Vec(((2*r+1)/3 - sqrt(r*(r+2)/3))/(2*x));
};
seq(18)
\\ test: y = 'x*Ser(seq(300), 'x); 0 == 9*x^3*y^4 - 6*x^2*y^3 + 2*x*(21*x - 1)*y^2 + (10*x - 1)*y + x
(PARI)
b(n) = sum(k=0, n\2, n!/(k!^2 * (n - 2*k)!)); \\ A002426
a(n) = 2*sum(k=0, n-1, binomial(2*n, k) * 3^k * b(n-k))/(n+1);
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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Reference gives 20 terms
Description corrected May 15 1997, thanks to Jean-Francois Beraud
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005
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STATUS
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approved
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