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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.

%I A091665 #20 Mar 30 2021 01:26:48
%S A091665 1,2,2,7,8,3,30,34,21,4,143,160,114,44,5,728,806,609,308,80,6,3876,
%T A091665 4256,3315,1908,715,132,7,21318,23256,18444,11420,5185,1482,203,8,
%U A091665 120175,130416,104652,67856,34520,12600,2814,296,9,690690,746350,603801,404016,221300,93924,27965,4984,414,10
%N 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.
%C A091665 Table II in the Brown reference.
%H A091665 Andrew Howroyd, <a href="/A091665/b091665.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A091665 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 A091665 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.
%e A091665 Triangle begins:
%e A091665     1;
%e A091665     2,   2;
%e A091665     7,   8,   3;
%e A091665    30,  34,  21,  4;
%e A091665   143, 160, 114, 44, 5;
%e A091665   ...
%e A091665 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
%p A091665 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);
%t A091665 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 *)
%o A091665 (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)!}
%o A091665 for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ _Andrew Howroyd_, Mar 29 2021
%Y A091665 Column 1 gives A006013, column 2 gives A046649, row sums give A000305.
%Y A091665 Same as A046652 but with rows reversed.
%Y A091665 Cf. A046653, A091599.
%K A091665 nonn,tabl
%O A091665 1,2
%A A091665 _Emeric Deutsch_, Mar 03 2004
%E A091665 Name clarified by _Andrew Howroyd_, Mar 29 2021