A342985 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.

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, 0, 7, 798, 7350, 7350, 798, 7, 0, 0, 8, 1568, 28400, 73700, 28400, 1568, 8, 0, 0, 9, 2826, 89073, 474588, 474588, 89073, 2826, 9, 0, 0, 10, 4770, 241220, 2292790, 4818092, 2292790, 241220, 4770, 10, 0

Offset: 0

Author

Andrew Howroyd, Apr 03 2021

Keywords

• nonn
• tabl

Comments

The number of vertices is n + 2 - k.

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

Examples

			Triangle begins:
  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;
  0, 7,  798,  7350,  7350,   798,    7, 0;
  0, 8, 1568, 28400, 73700, 28400, 1568, 8, 0;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
• T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VIIIa.

Crossrefs

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

Row sums are A342986.

Cf. A342980, A342982, A342984, A342987.

Programs

PARI

\\ here G(n,y) is A342984 as g.f.
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)}
G(n,y)={my(g=F(n,y)); subst(g, x, serreverse(x*g^2))}
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))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

Formula

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

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.