A104742 Number of rooted maps of (orientable) genus 3 containing n edges.

1485, 113256, 5008230, 167808024, 4721384790, 117593590752, 2675326679856, 56740864304592, 1137757854901806, 21789659909226960, 401602392805341924, 7165100439281414160, 124314235272290304540, 2105172926498512761984, 34899691847703927826500, 567797719808735191344672, 9084445205688065541367710

Offset: 6

Author

Valery A. Liskovets, Mar 22 2005

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 6..30 (from Mednykh and Nedela)
• E. A. Bender and E. R. Canfield, The number of rooted maps on an orientable surface, J. Combin. Theory, B 53 (1991), 293-299.
• Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
• S. R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
• A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, preprint (submitted to J. Combin. Th. B).

Crossrefs

Column k=3 of A238396.

Cf. A104596 (unrooted maps).

Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, this sequence, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

Programs

Mathematica

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 3];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, Jul 20 2018 *)

PARI

A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A104742_ser(N) = {
  my(y=A005159_ser(N+1));
  y*(y-1)^6*(460*y^8 - 3680*y^7 + 63055*y^6 - 198110*y^5 + 835954*y^4 - 1408808*y^3 + 1986832*y^2 - 1462400*y + 547552)/(81*(y-2)^12*(y+2)^7)
};
Vec(A104742_ser(17))  \\ Gheorghe Coserea, Jun 02 2017