A007137 Number of rooted maps with n edges on the projective plane. (Formerly M4734)

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

Offset

1,2

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

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)

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 *)

Prog

(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

Crossrefs

Cf. A006300.

A column of A267180.

Sequence in context: A190869 A254599 A217634 * A361137 A135927 A299952

Adjacent sequences: A007134 A007135 A007136 * A007138 A007139 A007140

Keyword

nonn,nice

Author

N. J. A. Sloane

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

Status

approved