A215402 Number of rooted maps of (orientable) genus 4 containing n edges.

225225, 24635754, 1495900107, 66519597474, 2416610807964, 75981252764664, 2141204115631518, 55352670009315660, 1334226671709010578, 30347730709395639732, 657304672067357799042, 13652607304062788395788, 273469313030628783700080, 5306599156694095573465824, 100128328831437989131706976, 1842794650155970906232185656

Offset: 8

Author

Alain Giorgetti, Aug 09 2012

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 8..500
• Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
• Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
• Alexander Mednykh, Alain Giorgetti, Enumeration of genus four maps by number of edges, Ars Mathematica Contemporanea 4 (2011), 351--361.

Crossrefs

Row sums of A269924.

Column g=4 of A269919.

Cf. A215019 (unrooted sensed maps), A297880 (unrooted unsensed maps).

Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, A104742, this sequence, 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, 4];
Table[a[n], {n, 8, 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);
A215402_ser(N) = {
  my(y=A005159_ser(N+1));
  -y*(y-1)^8*(15812*y^12 - 189744*y^11 + 4708549*y^10 - 24892936*y^9 + 173908449*y^8 - 567987942*y^7 + 1743939189*y^6 - 3485359548*y^5 + 5448471852*y^4 - 6051484928*y^3 + 4633500336*y^2 - 2228416192*y + 517976128)/(81*(y-2)^17*(y+2)^10);
};
Vec(A215402_ser(16)) \\ Gheorghe Coserea, Jun 02 2017

Extensions

More terms from Joerg Arndt, Feb 26 2014