cp's OEIS Frontend

A342989 Triangle read by rows: T(n,k) is the number of nonseparable rooted toroidal maps with n edges and k faces, n >= 2, k = 1..n-1.

%I A342989 #16 Jan 02 2022 16:09:19
%S A342989 1,4,4,10,39,10,20,190,190,20,35,651,1568,651,35,56,1792,8344,8344,
%T A342989 1792,56,84,4242,33580,64667,33580,4242,84,120,8988,111100,361884,
%U A342989 361884,111100,8988,120,165,17490,317680,1607125,2713561,1607125,317680,17490,165
%N A342989 Triangle read by rows: T(n,k) is the number of nonseparable rooted toroidal maps with n edges and k faces, n >= 2, k = 1..n-1.
%C A342989 The number of vertices is n - k.
%C A342989 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 A342989 Andrew Howroyd, <a href="/A342989/b342989.txt">Table of n, a(n) for n = 2..1276</a> (first 50 rows)
%H A342989 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 II.
%F A342989 T(n,n-k) = T(n,k).
%e A342989 Triangle begins:
%e A342989     1;
%e A342989     4,    4;
%e A342989    10,   39,     10;
%e A342989    20,  190,    190,     20;
%e A342989    35,  651,   1568,    651,     35;
%e A342989    56, 1792,   8344,   8344,   1792,     56;
%e A342989    84, 4242,  33580,  64667,  33580,   4242,   84;
%e A342989   120, 8988, 111100, 361884, 361884, 111100, 8988, 120;
%e A342989   ...
%o A342989 (PARI)
%o A342989 MQ(n,g,x=1)={ \\ after Quadric in A269921.
%o A342989   my(Q=matrix(n+1,g+1)); Q[1,1]=x;
%o A342989   for(n=1, n, for(g=0, min(n\2,g),
%o A342989      my(t = (1+x)*(2*n-1)/3 * Q[n, 1+g]
%o A342989        + if(g && n>1, (2*n-3)*(2*n-2)*(2*n-1)/12 * Q[n-1, g])
%o A342989        + sum(k = 1, n-1, sum(i = 0, g, (2*k-1) * (2*(n-k)-1) * Q[k, 1+i] * Q[n-k, 1+g-i]))/2);
%o A342989      Q[1+n, 1+g] = t * 6/(n+1); ));
%o A342989   Q
%o A342989 }
%o A342989 F(n,m,y,z)={my(Q=MQ(n,m,z)); sum(n=0, n, x^n*Ser(Q[1+n,]/z, y)) + O(x*x^n)}
%o A342989 H(n,g=1)={my(p=F(n,g,'y,'z), v=Vec(polcoef(subst(p, x, serreverse(x*p^2)), g, 'y))); vector(#v, n, Vecrev(v[n], n))}
%o A342989 { my(T=H(10)); for(n=1, #T, print(T[n])) }
%Y A342989 Columns 1..4 are A000292, A006408, A006409, A006410.
%Y A342989 Row sums are A343089.
%Y A342989 Cf. A082680 (planar case), A269921 (rooted toroidal maps), A343090, A343092.
%K A342989 nonn,tabl
%O A342989 2,2
%A A342989 _Andrew Howroyd_, Apr 04 2021