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
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Mar 03 2004
EXTENSIONS
Name clarified by Andrew Howroyd, Mar 29 2021
STATUS
approved