A007137 Number of rooted maps with n edges on the projective plane.

1, 10, 98, 982, 10062, 105024, 1112757, 11934910, 129307100, 1412855500, 15548498902, 172168201088, 1916619748084, 21436209373224, 240741065193282, 2713584138389838, 30687358107371442, 348061628432108352

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

References

• 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).

Links

• T. D. Noe, Table of n, a(n) for n = 1..100
• E. A. Bender, E. R. Canfield and R. W. Robinson, The enumeration of maps on the torus and the projective plane, Canad. Math. Bull., 31 (1988), 257-271; see p. 270.
• Guillaume Chapuy, Maciej Dołęga, A bijection for rooted maps on general surfaces, arXiv:1501.06942 [math.CO], 2016; see corollary 4.5.
• Valery A. Liskovets, A reductive technique for enumerating non-isomorphic planar maps, Discrete Math. 156 (1996), no. 1-3, 197--217. MR1405018 (97f:05087). - _N. J. A. Sloane_, Jun 03 2012

Crossrefs

Cf. A006300.

A column of A267180.

Programs

Maple

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

Mathematica

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 *)

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
\\ Gheorghe Coserea, Jul 07 2018

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);
vector(18, n, a(n)) \\ Gheorghe Coserea, Dec 26 2018

Formula

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)

From Gheorghe Coserea, Dec 26 2018: (Start)

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)

Extensions

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