A342987 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

1, 0, 1, 0, 2, 2, 0, 3, 15, 5, 0, 4, 60, 84, 14, 0, 5, 175, 650, 420, 42, 0, 6, 420, 3324, 5352, 1980, 132, 0, 7, 882, 13020, 42469, 37681, 9009, 429, 0, 8, 1680, 42240, 246540, 429120, 239752, 40040, 1430, 0, 9, 2970, 118998, 1142622, 3462354, 3711027, 1421226, 175032, 4862

Offset

0,5

Comments

The number of vertices is n + 2 - k.

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 VIIIb.

Formula

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

Example

Triangle begins:
  1;
  0, 1;
  0, 2,    2;
  0, 3,   15,     5;
  0, 4,   60,    84,     14;
  0, 5,  175,   650,    420,     42;
  0, 6,  420,  3324,   5352,   1980,    132;
  0, 7,  882, 13020,  42469,  37681,   9009,   429;
  0, 8, 1680, 42240, 246540, 429120, 239752, 40040, 1430;
  ...

Prog

(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, v=Vec(sqrt(serreverse(x/g^2)/x))); [Vecrev(t) | t<-v]}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

Crossrefs

Columns k=1..4 are A000007, A000027, A006470, A006471.

Diagonals are A000108, A002740, A006432, A006433.

Row sums are A342988.

Cf. A342981, A342982, A342984, A342985.

Sequence in context: A361893 A244129 A363907 * A090657 A167001 A108563

Adjacent sequences: A342984 A342985 A342986 * A342988 A342989 A342990

Keyword

nonn,tabl

Author

Andrew Howroyd, Apr 03 2021

Status

approved