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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).

%I A091599 #12 Mar 30 2021 01:26:25
%S A091599 1,2,1,6,6,1,24,26,12,1,110,120,75,20,1,546,594,416,174,30,1,2856,
%T A091599 3094,2289,1176,350,42,1,15504,16728,12768,7322,2880,636,56,1,86526,
%U A091599 93024,72420,44388,20475,6324,1071,72,1,493350,528770,417240,267240,136252,51495,12740,1700,90,1
%N 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).
%C A091599 Table I in the Brown reference.
%H A091599 Andrew Howroyd, <a href="/A091599/b091599.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A091599 W. G. Brown, <a href="http://dx.doi.org/10.4153/CJM-1963-056-7">Enumeration of non-separable planar maps</a>, Canad. J. Math., 15 (1963), 526-545.
%F A091599 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)!
%e A091599 Triangle starts:
%e A091599     1;
%e A091599     2,   1;
%e A091599     6,   6,  1;
%e A091599    24,  26, 12,  1;
%e A091599   110, 120, 75, 20, 1;
%e A091599   ...
%p A091599 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);
%o A091599 (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)!
%o A091599 for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ _Andrew Howroyd_, Mar 29 2021
%Y A091599 Column 1 gives A046646, column 2 gives A046647, row sums give A000259.
%Y A091599 Same as A046651 but with rows reversed.
%Y A091599 Cf. A046653, A091665.
%K A091599 nonn,tabl
%O A091599 1,2
%A A091599 _Emeric Deutsch_, Mar 03 2004
%E A091599 Name clarified by _Andrew Howroyd_, Mar 29 2021