%I #11 Apr 15 2021 11:53:02

%S 1,4,10,10,79,70,20,340,900,420,35,1071,5846,7885,2310,56,2772,26320,

%T 71372,59080,12012,84,6258,93436,431739,706068,398846,60060,120,12768,

%U 280120,2000280,5494896,6052840,2499096,291720,165,24090,739420,7643265,32055391,58677420,46759630,14805705,1385670

%N 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 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="/A343092/b343092.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 VId.

%e Triangle begins:

%e 1;

%e 4, 10;

%e 10, 79, 70;

%e 20, 340, 900, 420;

%e 35, 1071, 5846, 7885, 2310;

%e 56, 2772, 26320, 71372, 59080, 12012;

%e 84, 6258, 93436, 431739, 706068, 398846, 60060;

%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, 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..2 are A000292, A006469.

%Y Diagonals are A002802, A006425, A006426, A006427.

%Y Row sums are A343093.

%Y Cf. A269921, A342981, A342989, A343090.

%K nonn,tabl

%O 2,2

%A _Andrew Howroyd_, Apr 04 2021