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A006295 Number of genus 1 rooted maps with 2 faces with n vertices.

%I A006295 M4739 #37 Jan 18 2025 09:08:05
%S A006295 10,167,1720,14065,100156,649950,3944928,22764165,126264820,678405090,
%T A006295 3550829360,18182708362,91392185080,452077562620,2205359390592,
%U A006295 10627956019245,50668344988068,239250231713210,1120028580999440,5202779260636958,23998704563581000,109991785264412452
%N A006295 Number of genus 1 rooted maps with 2 faces with n vertices.
%D A006295 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006295 T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.
%H A006295 Vincenzo Librandi, <a href="/A006295/b006295.txt">Table of n, a(n) for n = 3..1000</a>
%H A006295 W. T. Tutte, <a href="http://dx.doi.org/10.1090/S0002-9904-1968-11877-4">On the enumeration of planar maps</a>, Bull. Amer. Math. Soc. 74 1968 64-74.
%H A006295 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.
%H A006295 <a href="/A007401/a007401_1.pdf">Notes</a>
%F A006295 G.f.: x*(1-sqrt(1-4*x))*(11+12*x+9*sqrt(1-4*x))/(4*(4*x-1)^4). - _Sean A. Irvine_, Nov 14 2010
%t A006295 Rest[CoefficientList[Series[(1 - Sqrt[1 - 4 x]) (11 + 12 x + 9 Sqrt[1 - 4 x]) / (4 (4 x - 1)^4), {x, 0, 40}], x]] (* _Vincenzo Librandi_, Jun 06 2017 *)
%o A006295 (PARI)
%o A006295 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A006295 A006295_ser(N) = {
%o A006295   my(y = A000108_ser(N+1));  y*(y-1)^3*(y^2 + 15*y - 6)/(y-2)^8;
%o A006295 };
%o A006295 Vec(A006295_ser(31)) \\ _Gheorghe Coserea_, Jun 04 2017
%o A006295 (PARI) my(x = 'x + O('x^60)); Vec(x*(1-sqrt(1-4*x))*(11+12*x+9*sqrt(1-4*x))/(4*(4*x-1)^4)) \\ _Michel Marcus_, Jun 05 2017
%Y A006295 Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, this sequence, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
%Y A006295 Column 2 of A269921, column 1 of A270406.
%K A006295 nonn
%O A006295 3,1
%A A006295 _N. J. A. Sloane_
%E A006295 More terms from _Sean A. Irvine_, Nov 14 2010