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%I #10 Apr 15 2021 11:51:29
%S 1,4,4,10,47,10,20,240,240,20,35,831,2246,831,35,56,2282,12656,12656,
%T 2282,56,84,5362,52164,109075,52164,5362,84,120,11256,173776,648792,
%U 648792,173776,11256,120,165,21690,495820,2978245,5360286,2978245,495820,21690,165
%N Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without separating cycles or isthmuses, n >= 2, k = 1..n-1.
%C The number of vertices is n-k.
%C 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 Andrew Howroyd, <a href="/A343090/b343090.txt">Table of n, a(n) for n = 2..1276</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, Table VIc.
%F T(n,n-k) = T(n,k).
%e Triangle begins:
%e 1;
%e 4, 4;
%e 10, 47, 10;
%e 20, 240, 240, 20;
%e 35, 831, 2246, 831, 35;
%e 56, 2282, 12656, 12656, 2282, 56;
%e 84, 5362, 52164, 109075, 52164, 5362, 84;
%e 120, 11256, 173776, 648792, 648792, 173776, 11256, 120;
%e ...
%o (PARI) \\ Needs F from A342989.
%o G(n,m,y,z)={my(p=F(n,m,y,z)); subst(p, x, serreverse(x*p^2))}
%o H(n, g=1)={my(q=G(n, g, 'y, 'z)-x*(1+'z), v=Vec(polcoef(sqrt(serreverse(x/q^2)/x), g, 'y))); [Vecrev(t) | t<-v]}
%o { my(T=H(10)); for(n=1, #T, print(T[n])) }
%Y Columns 1..4 are A000292, A006422, A006423, A006424.
%Y Row sums are A343091.
%Y Cf. A269921, A342980, A342989, A343092.
%K nonn,tabl
%O 2,2
%A _Andrew Howroyd_, Apr 04 2021
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