A342988 -id:A342988 - cp's OEIS frontend

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

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 2 - k.

For k >= 2, column k is a polynomial of degree 4*(k-2)+1.

			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;
  ...

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.

Columns k=1..4 are A000007, A000027, A006470, A006471.
Diagonals are A000108, A002740, A006432, A006433.
Row sums are A342988.
Cf. A342981, A342982, A342984, A342985.

\\ 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])) }

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.

A342986 Number of tree-rooted loopless planar maps with n edges and no isthmuses.

Original entry on oeis.org

1, 0, 2, 6, 44, 280, 2100, 16310, 133652, 1132992, 9895672, 88520520, 808057712, 7504219008, 70730676392, 675328163542, 6521495669380, 63612394972608, 626076210568200, 6211621325369992, 62077602307372720, 624488579671582880, 6320044589443116720, 64313288809475362888

Offset: 0

Andrew Howroyd, Apr 03 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Row sums of A342985.

Cf. A004304, A342988.

\\ here J(n) gives A004304 as g.f.
J(n)={my(g=(1-sqrt(1-4*x+O(x^3*x^n)))/(2*x), h=serconvol(g, (g-1)/x));sqrt(x/serreverse(x*h^2))}
seq(n)={my(g=J(n)-2*x, p=O(1)); while(serprec(p, x)<=n, p = subst(g, x, x*p^2)); Vec(p)}

G.f.: A(x) satisfies A(x) = G(x*A(x)^2) where G(x) + 2*x is the g.f. of A004304.

cp's OEIS Frontend

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.

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