%I #14 Apr 06 2021 21:13:29

%S 1,0,0,0,2,0,0,3,3,0,0,4,36,4,0,0,5,135,135,5,0,0,6,360,1368,360,6,0,

%T 0,7,798,7350,7350,798,7,0,0,8,1568,28400,73700,28400,1568,8,0,0,9,

%U 2826,89073,474588,474588,89073,2826,9,0,0,10,4770,241220,2292790,4818092,2292790,241220,4770,10,0

%N Triangle read by rows: T(n,k) is the number of tree-rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

%C The number of vertices is n + 2 - k.

%C For k >= 2, column k without the initial zero term is a polynomial of degree 4*(k-2)+1.

%H Andrew Howroyd, <a href="/A342985/b342985.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%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 VIIIa.

%F T(n,n+2-k) = T(n,k).

%F G.f.: A(x,y) satisfies A(x,y) = G(x*A(x,y)^2),y) where G(x,y) + x*(1+y) is the g.f. of A342984.

%e Triangle begins:

%e 1;

%e 0, 0;

%e 0, 2, 0;

%e 0, 3, 3, 0;

%e 0, 4, 36, 4, 0;

%e 0, 5, 135, 135, 5, 0;

%e 0, 6, 360, 1368, 360, 6, 0;

%e 0, 7, 798, 7350, 7350, 798, 7, 0;

%e 0, 8, 1568, 28400, 73700, 28400, 1568, 8, 0;

%e ...

%o (PARI) \\ here G(n,y) is A342984 as g.f.

%o 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 G(n,y)={my(g=F(n,y)); subst(g, x, serreverse(x*g^2))}

%o H(n)={my(g=G(n, y)-x*(1+y), v=Vec(sqrt(serreverse(x/g^2)/x))); vector(#v, n, Vecrev(v[n], n))}

%o { my(T=H(8)); for(n=1, #T, print(T[n])) }

%Y Columns and diagonals 3..5 are A006428, A006429, A006430.

%Y Row sums are A342986.

%Y Cf. A342980, A342982, A342984, A342987.

%K nonn,tabl

%O 0,5

%A _Andrew Howroyd_, Apr 03 2021