%I #30 Aug 23 2024 03:17:49
%S 225225,24635754,1495900107,66519597474,2416610807964,75981252764664,
%T 2141204115631518,55352670009315660,1334226671709010578,
%U 30347730709395639732,657304672067357799042,13652607304062788395788,273469313030628783700080,5306599156694095573465824,100128328831437989131706976,1842794650155970906232185656
%N Number of rooted maps of (orientable) genus 4 containing n edges.
%H Sean R. Carrell, Guillaume Chapuy, <a href="http://arxiv.org/abs/1402.6300">Simple recurrence formulas to count maps on orientable surfaces</a>, arXiv:1402.6300 [math.CO], (19-March-2014).
%H S. R. Finch, <a href="https://arxiv.org/abs/2408.12440">An exceptional convolutional recurrence</a>, arXiv:2408.12440 [math.CO], 22 Aug 2024.
%H Alexander Mednykh, Alain Giorgetti, <a href="http://amc-journal.eu/index.php/amc/article/view/132">Enumeration of genus four maps by number of edges</a>, Ars Mathematica Contemporanea 4 (2011), 351--361.
%t 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);
%t a[n_] := T[n, 4];
%t Table[a[n], {n, 8, 30}] (* _Jean-François Alcover_, Jul 20 2018 *)
%o (PARI)
%o A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
%o A215402_ser(N) = {
%o my(y=A005159_ser(N+1));
%o -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);
%o };
%o Vec(A215402_ser(16)) \\ _Gheorghe Coserea_, Jun 02 2017
%Y Cf. A215019 (unrooted maps).
%Y Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, A104742, this sequence, A238355, A238356, A238357, A238358, A238359, A238360.
%K nonn
%O 8,1
%A _Alain Giorgetti_, Aug 09 2012
%E More terms from _Joerg Arndt_, Feb 26 2014