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A006296 Number of genus 1 rooted maps with 3 faces with n vertices.

%I A006296 M5341 #34 Jul 08 2025 16:46:29
%S A006296 70,1720,24164,256116,2278660,17970784,129726760,875029804,5593305476,
%T A006296 34225196720,201976335288,1156128848680,6447533938280,35155923872640,
%U A006296 187959014565840,987658610225052,5110652802256260,26084524995672080,131501187454625560,655590388845975000,3235463376771463288,15820770680078552000,76708503479715247920,369046200766330733880,1762793459781859039080,8364468224596427692896,39445646133672676352560,184956513528952419546448,862615498961026097997392,4003067488703222112053760,18489846573354278755829152,85028133934182275077421180,389398354121840111751946628,1776360539933013004774353872,8073622060225813990245976280,36567311475673299914222851832
%N A006296 Number of genus 1 rooted maps with 3 faces with n vertices.
%D A006296 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006296 T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.
%H A006296 Vincenzo Librandi, <a href="/A006296/b006296.txt">Table of n, a(n) for n = 4..1000</a>
%H A006296 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 A006296 <a href="/A007401/a007401_1.pdf">Notes</a>
%F A006296 G.f.: x(1-sqrt(1-4*x))(45+152*x+(25+8*x)sqrt(1-4*x))/(2(1-4*x)^(11/2)). - _Sean A. Irvine_, Nov 14 2010
%t A006296 Rest[CoefficientList[Series[(1 - Sqrt[1 - 4 x]) (45 + 152 x + (25 + 8 x) Sqrt[1 - 4 x]) / (2 (1 - 4 x)^(11 / 2)), {x, 0, 40}], x]] (* _Vincenzo Librandi_, Jun 06 2017 *)
%o A006296 (PARI)
%o A006296 A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
%o A006296 A006296_ser(N) = {
%o A006296   my(y = A000108_ser(N+1));
%o A006296   -2*y*(y-1)^4*(10*y^3 + 97*y^2 - 64*y - 8)/(y-2)^11;
%o A006296 };
%o A006296 Vec(A006296_ser(36)) \\ _Gheorghe Coserea_, Jun 04 2017
%Y A006296 Rooted maps of genus 1 with n edges and f faces for 1<=f<=10: A002802(with offset 2) f=1, A006295 f=2, this sequence, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
%Y A006296 Column 3 of A269921, column g=1 of A270407.
%K A006296 nonn
%O A006296 4,1
%A A006296 _N. J. A. Sloane_
%E A006296 More terms from _Sean A. Irvine_, Nov 14 2010