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A006469 Number of rooted toroidal maps with 2 faces, n vertices and no isthmuses.

%I A006469 M4727 #36 Aug 20 2025 12:15:45
%S A006469 10,79,340,1071,2772,6258,12768,24090,42702,71929,116116,180817,
%T A006469 273000,401268,576096,810084,1118226,1518195,2030644,2679523,3492412,
%U A006469 4500870,5740800,7252830,9082710,11281725,13907124,17022565,20698576,25013032,30051648,35908488
%N A006469 Number of rooted toroidal maps with 2 faces, n vertices and no isthmuses.
%C A006469 A map on a torus has genus 1.
%D A006469 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006469 Colin Barker, <a href="/A006469/b006469.txt">Table of n, a(n) for n = 1..1000</a>
%H A006469 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 A006469 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A006469 G.f.: x/(x-1)^7*(3*x^2-9*x-10). - _Simon Plouffe_, Master's thesis, Uqam 1992
%F A006469 From _Colin Barker_, Apr 22 2017: (Start)
%F A006469 a(n) = (n*(474 + 1247*n + 1215*n^2 + 545*n^3 + 111*n^4 + 8*n^5)) / 360.
%F A006469 a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
%F A006469 (End)
%t A006469 A006469[n_] := n*(n + 1)*(n + 2)*(n + 3)*(n*(8*n + 63) + 79)/360;
%t A006469 Array[A006469, 50] (* _Paolo Xausa_, Aug 20 2025 *)
%o A006469 (PARI) Vec(x*(10 + 9*x - 3*x^2) / (1 - x)^7 + O(x^40)) \\ _Colin Barker_, Apr 22 2017
%Y A006469 Column 2 of A343092.
%K A006469 nonn,easy
%O A006469 1,1
%A A006469 _N. J. A. Sloane_
%E A006469 Name improved by _Sean A. Irvine_, Apr 21 2017