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Number of rooted toroidal maps with 2 faces and n vertices and without separating cycles or isthmuses.
(Formerly M3684)
5

%I M3684 #17 May 15 2023 17:26:54

%S 4,47,240,831,2282,5362,11256,21690,39072,66649,108680,170625,259350,

%T 383348,552976,780708,1081404,1472595,1974784,2611763,3410946,4403718,

%U 5625800,7117630,8924760,11098269,13695192,16778965,20419886,24695592,29691552,35501576

%N Number of rooted toroidal maps with 2 faces and n vertices and without separating cycles or isthmuses.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A006422/b006422.txt">Table of n, a(n) for n = 1..1000</a>

%H T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(75)90050-7">Counting rooted maps by genus. III: Nonseparable maps</a>, J. Combinatorial Theory Ser. B 18 (1975), 222-259.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F From _Colin Barker_, Apr 09 2013: (Start)

%F a(n) = n*(n + 1)*(n + 2)*(8*n^3 + 87*n^2 + 148*n - 3)/360.

%F G.f.: x*(2*x^3+5*x^2-19*x-4) / (x-1)^7. (End)

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{4,47,240,831,2282,5362,11256},40] (* _Harvey P. Dale_, May 15 2023 *)

%o (PARI) a(n) = {n*(n + 1)*(n + 2)*(8*n^3 + 87*n^2 + 148*n - 3)/360}

%Y Column 2 of A343090.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E Name clarified and terms a(11) and beyond from _Andrew Howroyd_, Apr 04 2021