A006295 Number of genus 1 rooted maps with 2 faces with n vertices (Formerly M4739)

10, 167, 1720, 14065, 100156, 649950, 3944928, 22764165, 126264820, 678405090, 3550829360, 18182708362, 91392185080, 452077562620, 2205359390592, 10627956019245, 50668344988068, 239250231713210, 1120028580999440, 5202779260636958, 23998704563581000, 109991785264412452, 501177338394193600, 2271383798432172450, 10243348296746854536, 45984242301056376660, 205559341637958057568, 915285296612144724660, 4060537494602836978160, 17952439453856255549176, 79117286081659327659520

Offset

3,1

References

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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 3..1000

W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 1968 64-74.

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.

Notes

Formula

G.f.: x(1-sqrt(1-4*x))(11+12*x+9*sqrt(1-4*x))/(4(4*x-1)^4). - Sean A. Irvine, Nov 14 2010

Mathematica

Rest[CoefficientList[Series[(1 - Sqrt[1 - 4 x]) (11 + 12 x + 9 Sqrt[1 - 4 x]) / (4 (4 x - 1)^4), {x, 0, 40}], x]] (* Vincenzo Librandi, Jun 06 2017 *)

Prog

(PARI)
A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A006295_ser(N) = {
  my(y = A000108_ser(N+1));  y*(y-1)^3*(y^2 + 15*y - 6)/(y-2)^8;
};
Vec(A006295_ser(31)) \\ Gheorghe Coserea, Jun 04 2017
(PARI) x = 'x + O('x^60); Vec(x*(1-sqrt(1-4*x))*(11+12*x+9*sqrt(1-4*x))/(4*(4*x-1)^4)) \\ Michel Marcus, Jun 05 2017

Crossrefs

Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, this sequence, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.

Column 2 of A269921, column 1 of A270406.

Sequence in context: A054688 A229228 A112650 * A006297 A279292 A099711

Adjacent sequences: A006292 A006293 A006294 * A006296 A006297 A006298

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

More terms from Sean A. Irvine, Nov 14 2010

Status

approved