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%I M5120 #35 Mar 28 2021 19:31:43
%S 0,0,0,0,21,966,27954,650076,13271982,248371380,4366441128,
%T 73231116024,1183803697278,18579191525700,284601154513452,
%U 4272100949982600,63034617139799916,916440476048146056,13154166812674577412,186700695099591735024,2623742783421329300190,36548087103760045010148,505099724454854883618924
%N Number of rooted genus-2 maps with n edges.
%D E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.
%H T. D. Noe, <a href="/A006301/b006301.txt">Table of n, a(n) for n=0..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 T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(72)90056-1">Counting rooted maps by genus</a>, J. Comb. Thy B13 (1972), 122-141 and 192-218.
%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, 2];
%t Table[a[n], {n, 0, 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 A006301_ser(N) = {
%o my(y=A005159_ser(N+1));
%o -y*(y-1)^4*(4*y^4 - 16*y^3 + 153*y^2 - 148*y + 196)/(9*(y-2)^7*(y+2)^4);
%o };
%o concat([0,0,0,0], Vec(A006301_ser(19))) \\ _Gheorghe Coserea_, Jun 02 2017
%Y Column k=2 of A238396.
%Y Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, this sequence, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.
%K nonn
%O 0,5
%A _N. J. A. Sloane_ and _Simon Plouffe_
%E More terms from _Joerg Arndt_, Feb 26 2014
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