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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. 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The number of vertices is n + 2 - k.

A tree-rooted planar map is a planar map with a distinguished spanning tree.

For k >= 2, column k is a polynomial of degree 4*(k-2)+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), 222-259, Table IVa.

FORMULA

T(n,n+2-k) = 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=n-i); 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

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Last modified December 7 13:02 EST 2022. Contains 358656 sequences. (Running on oeis4.)