A091599 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with r*n edges and a fixed outer face of r*k edges which are invariant under a rotation of 1/r for any r >= 2 (independent of actual value of r).

1, 2, 1, 6, 6, 1, 24, 26, 12, 1, 110, 120, 75, 20, 1, 546, 594, 416, 174, 30, 1, 2856, 3094, 2289, 1176, 350, 42, 1, 15504, 16728, 12768, 7322, 2880, 636, 56, 1, 86526, 93024, 72420, 44388, 20475, 6324, 1071, 72, 1, 493350, 528770, 417240, 267240, 136252, 51495, 12740, 1700, 90, 1

Offset

1,2

Comments

Table I in the Brown reference.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

Formula

T(n, k) = k*(Sum_{j=k..min(n, 2*k)} (2*j-k)*(j-1)!*(3*n-j-k-1)!/(((j-k)!)^2*(2*k-j)!*(n-j)!))/(2*n-k)!

Example

Triangle starts:
    1;
    2,   1;
    6,   6,  1;
   24,  26, 12,  1;
  110, 120, 75, 20, 1;
  ...

Maple

T := proc(n, k) if k<=n then k*sum((2*j-k)*(j-1)!*(3*n-j-k-1)!/(j-k)!/(j-k)!/(2*k-j)!/(n-j)!, j=k..min(n, 2*k))/(2*n-k)! else 0 fi end: seq(seq(T(n, k), k=1..n), n=1..11);

Prog

(PARI) T(n, k) = k*sum(j=k, min(n, 2*k), (2*j-k)*(j-1)!*(3*n-j-k-1)!/(((j-k)!)^2*(2*k-j)!*(n-j)!))/(2*n-k)!
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Mar 29 2021

Crossrefs

Column 1 gives A046646, column 2 gives A046647, row sums give A000259.

Same as A046651 but with rows reversed.

Cf. A046653, A091665.

Sequence in context: A244888 A130561 A157400 * A048999 A066667 A105278

Adjacent sequences: A091596 A091597 A091598 * A091600 A091601 A091602

Keyword

nonn,tabl

Author

Emeric Deutsch, Mar 03 2004

Extensions

Name clarified by Andrew Howroyd, Mar 29 2021

Status

approved