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
KEYWORD
nonn,nice
AUTHOR
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