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A342984 Triangle read by rows: T(n,k) is the number of nonseparable tree-rooted planar maps with n edges and k faces, n >= 0, k = 1..n+1.

%I A342984 #12 Jan 02 2022 16:09:36
%S A342984 1,1,1,0,2,0,0,3,3,0,0,4,20,4,0,0,5,75,75,5,0,0,6,210,604,210,6,0,0,7,
%T A342984 490,3150,3150,490,7,0,0,8,1008,12480,27556,12480,1008,8,0,0,9,1890,
%U A342984 40788,170793,170793,40788,1890,9,0,0,10,3300,115500,829920,1565844,829920,115500,3300,10,0
%N A342984 Triangle read by rows: T(n,k) is the number of nonseparable tree-rooted planar maps with n edges and k faces, n >= 0, k = 1..n+1.
%C A342984 The number of vertices is n + 2 - k.
%C A342984 A tree-rooted planar map is a planar map with a distinguished spanning tree.
%C A342984 For k >= 2, column k is a polynomial of degree 4*(k-2)+1.
%H A342984 Andrew Howroyd, <a href="/A342984/b342984.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A342984 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 IVa.
%F A342984 T(n,n+2-k) = T(n,k).
%F A342984 G.f. A(x,y) satisfies F(x,y) = A(x*F(x,y)^2,y) where F(x,y) is the g.f. of A342982.
%e A342984 Triangle begins:
%e A342984   1;
%e A342984   1, 1;
%e A342984   0, 2,    0;
%e A342984   0, 3,    3,     0;
%e A342984   0, 4,   20,     4,     0;
%e A342984   0, 5,   75,    75,     5,     0;
%e A342984   0, 6,  210,   604,   210,     6,    0;
%e A342984   0, 7,  490,  3150,  3150,   490,    7, 0;
%e A342984   0, 8, 1008, 12480, 27556, 12480, 1008, 8, 0;
%e A342984   ...
%o A342984 (PARI) \\ here F(n,y) gives A342982 as g.f.
%o A342984 F(n,y)={sum(n=0, n, x^n*sum(i=0, n, my(j=n-i); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)}
%o A342984 H(n)={my(g=F(n,y), v=Vec(subst(g, x, serreverse(x*g^2)))); vector(#v, n, Vecrev(v[n], n))}
%o A342984 { my(T=H(8)); for(n=1, #T, print(T[n])) }
%Y A342984 Columns (and diagonals) 3..5 are A006411, A006412, A006413.
%Y A342984 Row sums are A004304.
%Y A342984 Cf. A342982, A342985, A342987.
%K A342984 nonn,tabl
%O A342984 0,5
%A A342984 _Andrew Howroyd_, Apr 03 2021