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A343090 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.

%I A343090 #10 Apr 15 2021 11:51:29
%S A343090 1,4,4,10,47,10,20,240,240,20,35,831,2246,831,35,56,2282,12656,12656,
%T A343090 2282,56,84,5362,52164,109075,52164,5362,84,120,11256,173776,648792,
%U A343090 648792,173776,11256,120,165,21690,495820,2978245,5360286,2978245,495820,21690,165
%N A343090 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 A343090 The number of vertices is n-k.
%C A343090 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 A343090 Andrew Howroyd, <a href="/A343090/b343090.txt">Table of n, a(n) for n = 2..1276</a>
%H A343090 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 A343090 T(n,n-k) = T(n,k).
%e A343090 Triangle begins:
%e A343090     1;
%e A343090     4,     4;
%e A343090    10,    47,     10;
%e A343090    20,   240,    240,     20;
%e A343090    35,   831,   2246,    831,     35;
%e A343090    56,  2282,  12656,  12656,   2282,     56;
%e A343090    84,  5362,  52164, 109075,  52164,   5362,    84;
%e A343090   120, 11256, 173776, 648792, 648792, 173776, 11256, 120;
%e A343090   ...
%o A343090 (PARI) \\ Needs F from A342989.
%o A343090 G(n,m,y,z)={my(p=F(n,m,y,z)); subst(p, x, serreverse(x*p^2))}
%o A343090 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 A343090 { my(T=H(10)); for(n=1, #T, print(T[n])) }
%Y A343090 Columns 1..4 are A000292, A006422, A006423, A006424.
%Y A343090 Row sums are A343091.
%Y A343090 Cf. A269921, A342980, A342989, A343092.
%K A343090 nonn,tabl
%O A343090 2,2
%A A343090 _Andrew Howroyd_, Apr 04 2021