

A342984


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


7



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, 490, 3150, 3150, 490, 7, 0, 0, 8, 1008, 12480, 27556, 12480, 1008, 8, 0, 0, 9, 1890, 40788, 170793, 170793, 40788, 1890, 9, 0, 0, 10, 3300, 115500, 829920, 1565844, 829920, 115500, 3300, 10, 0
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OFFSET

0,5


COMMENTS

The number of vertices is n + 2  k.
A treerooted planar map is a planar map with a distinguished spanning tree.
For k >= 2, column k is a polynomial of degree 4*(k2)+1.


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), 222259, Table IVa.


FORMULA

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


EXAMPLE

Triangle begins:
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, 490, 3150, 3150, 490, 7, 0;
0, 8, 1008, 12480, 27556, 12480, 1008, 8, 0;
...


PROG

(PARI) \\ here F(n, y) gives A342982 as g.f.
F(n, y)={sum(n=0, n, x^n*sum(i=0, n, my(j=ni); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)}
H(n)={my(g=F(n, y), v=Vec(subst(g, x, serreverse(x*g^2)))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }


CROSSREFS

Columns (and diagonals) 3..5 are A006411, A006412, A006413.
Row sums are A004304.
Cf. A342982, A342985, A342987.
Sequence in context: A341841 A050186 A334218 * A342985 A278094 A245487
Adjacent sequences: A342981 A342982 A342983 * A342985 A342986 A342987


KEYWORD

nonn,tabl


AUTHOR

Andrew Howroyd, Apr 03 2021


STATUS

approved



