A091665 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with 2*n+1 edges and a fixed outer face of 2*k edges which are invariant under a rotation of a 1/2 turn.

1, 2, 2, 7, 8, 3, 30, 34, 21, 4, 143, 160, 114, 44, 5, 728, 806, 609, 308, 80, 6, 3876, 4256, 3315, 1908, 715, 132, 7, 21318, 23256, 18444, 11420, 5185, 1482, 203, 8, 120175, 130416, 104652, 67856, 34520, 12600, 2814, 296, 9, 690690, 746350, 603801, 404016, 221300, 93924, 27965, 4984, 414, 10

Offset

1,2

Comments

Table II 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-1)} (2*j-k+1)*(j-1)!*(3*n-k-j)!/((j-k+1)!*(j-k)!*(2*k-j-1)!*(n-j)!))/(2*n-k+1)! for k<=n and T(n, k)=0 for k>n.

Example

Triangle begins:
    1;
    2,   2;
    7,   8,   3;
   30,  34,  21,  4;
  143, 160, 114, 44, 5;
  ...
The T(n,n) = n solutions correspond to a regular polygon with 2n vertices and a single diagonal joining two diametrically opposite vertices. - Andrew Howroyd, Mar 29 2021

Maple

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

Mathematica

t[n_, k_] := If[k <= n, k*Sum[(2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!, {j, k, Min[n, 2*k-1]}]/(2*n-k+1)!, 0]; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 20 2014, after Maple *)

Prog

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

Crossrefs

Column 1 gives A006013, column 2 gives A046649, row sums give A000305.

Same as A046652 but with rows reversed.

Cf. A046653, A091599.

Sequence in context: A263868 A263793 A232647 * A019905 A230384 A151864

Adjacent sequences: A091662 A091663 A091664 * A091666 A091667 A091668

Keyword

nonn,tabl

Author

Emeric Deutsch, Mar 03 2004

Extensions

Name clarified by Andrew Howroyd, Mar 29 2021

Status

approved