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A343092 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without isthmuses, n >= 2, k = 1..n-1.

%I A343092 #11 Apr 15 2021 11:53:02
%S A343092 1,4,10,10,79,70,20,340,900,420,35,1071,5846,7885,2310,56,2772,26320,
%T A343092 71372,59080,12012,84,6258,93436,431739,706068,398846,60060,120,12768,
%U A343092 280120,2000280,5494896,6052840,2499096,291720,165,24090,739420,7643265,32055391,58677420,46759630,14805705,1385670
%N A343092 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without isthmuses, n >= 2, k = 1..n-1.
%C A343092 The number of vertices is n - k.
%C A343092 Column k is a polynomial of degree 3*k. This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.
%H A343092 Andrew Howroyd, <a href="/A343092/b343092.txt">Table of n, a(n) for n = 2..1276</a>
%H A343092 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, Table VId.
%e A343092 Triangle begins:
%e A343092    1;
%e A343092    4,   10;
%e A343092   10,   79,    70;
%e A343092   20,  340,   900,    420;
%e A343092   35, 1071,  5846,   7885,   2310;
%e A343092   56, 2772, 26320,  71372,  59080,  12012;
%e A343092   84, 6258, 93436, 431739, 706068, 398846, 60060;
%e A343092   ...
%o A343092 (PARI) \\ Needs F from A342989.
%o A343092 G(n,m,y,z)={my(p=F(n,m,y,z)); subst(p, x, serreverse(x*p^2))}
%o A343092 H(n, g=1)={my(q=G(n, g, 'y, 'z)-x, v=Vec(polcoef(sqrt(serreverse(x/q^2)/x), g, 'y))); [Vecrev(t) | t<-v]}
%o A343092 { my(T=H(10)); for(n=1, #T, print(T[n])) }
%Y A343092 Columns 1..2 are A000292, A006469.
%Y A343092 Diagonals are A002802, A006425, A006426, A006427.
%Y A343092 Row sums are A343093.
%Y A343092 Cf. A269921, A342981, A342989, A343090.
%K A343092 nonn,tabl
%O A343092 2,2
%A A343092 _Andrew Howroyd_, Apr 04 2021