A104742 - OEIS

%I #33 Aug 23 2024 03:17:21

%S 1485,113256,5008230,167808024,4721384790,117593590752,2675326679856,

%T 56740864304592,1137757854901806,21789659909226960,401602392805341924,

%U 7165100439281414160,124314235272290304540,2105172926498512761984,34899691847703927826500,567797719808735191344672,9084445205688065541367710

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

%H T. D. Noe, <a href="/A104742/b104742.txt">Table of n, a(n) for n = 6..30</a> (from Mednykh and Nedela)

%H E. A. Bender and E. R. Canfield, <a href="https://doi.org/10.1016/0095-8956(91)90079-Y">The number of rooted maps on an orientable surface</a>, J. Combin. Theory, B 53 (1991), 293-299.

%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 A. D. Mednykh and R. Nedela, <a href="http://www.savbb.sk/mu/articles/4_2004_nedela.pdf">Enumeration of unrooted maps with given genus</a>, preprint (submitted to J. Combin. Th. B).

%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, 3];

%t Table[a[n], {n, 6, 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 A104742_ser(N) = {

%o   my(y=A005159_ser(N+1));

%o   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)

%o };

%o Vec(A104742_ser(17))  \\ _Gheorghe Coserea_, Jun 02 2017

%Y Column k=3 of A238396.

%Y Cf. A104596 (unrooted maps).

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

%K nonn

%O 6,1

%A _Valery A. Liskovets_, Mar 22 2005