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A342061 Triangle read by rows: T(n,k) is the number of sensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.

%I A342061 #14 Jan 02 2022 16:09:59
%S A342061 1,1,1,1,1,1,1,2,2,1,1,3,8,3,1,1,4,16,16,4,1,1,5,38,63,38,5,1,1,7,72,
%T A342061 218,218,72,7,1,1,8,134,622,1075,622,134,8,1,1,10,224,1600,4214,4214,
%U A342061 1600,224,10,1,1,12,375,3703,14381,22222,14381,3703,375,12,1
%N A342061 Triangle read by rows: T(n,k) is the number of sensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.
%C A342061 The number of faces is n + 2 - k.
%H A342061 Andrew Howroyd, <a href="/A342061/b342061.txt">Table of n, a(n) for n = 2..1276</a> (first 50 rows)
%H A342061 Timothy R. Walsh, <a href="https://doi.org/10.1016/j.disc.2004.08.036">Efficient enumeration of sensed planar maps</a>, Discrete Math. 293 (2005), no. 1-3, 263--289. MR2136069 (2006b:05062).
%F A342061 T(n,k) = T(n, n+2-k).
%e A342061 Triangle begins:
%e A342061   1;
%e A342061   1, 1;
%e A342061   1, 1,   1;
%e A342061   1, 2,   2,   1;
%e A342061   1, 3,   8,   3,    1;
%e A342061   1, 4,  16,  16,    4,   1;
%e A342061   1, 5,  38,  63,   38,   5,   1;
%e A342061   1, 7,  72, 218,  218,  72,   7, 1;
%e A342061   1, 8, 134, 622, 1075, 622, 134, 8, 1;
%e A342061   ...
%o A342061 (PARI) \\ See section 4 of Walsh reference.
%o A342061 T(n)={
%o A342061   my(B=matrix(n, n, i, j, if(i+j <= n+1, (2*i+j-2)!*(2*j+i-2)!/(i!*j!*(2*i-1)!*(2*j-1)!))));
%o A342061   my(C(i,j)=((i+j-1)*(i+1)*(j+1)/(2*(2*i+j-1)*(2*j+i-1)))*B[(i+1)/2,(j+1)/2]);
%o A342061   my(D(i,j)=((j+1)/2)*B[i/2, (j+1)/2]);
%o A342061   my(E(i,j)=((i-1)*(j-1) + 2*(i+j)*(i+j-1))*B[i,j]);
%o A342061   my(F(i,j)=if(!i, j==1, ((i+j)*(6*j+2*i-5)*j*(2*i+j-1)/(2*(2*i+1)*(2*j+i-2)))*B[i,j]) + if(j-1, binomial(i+2,2)*B[i+1,j-1]));
%o A342061   vector(n, n, vector(n, i, my(j=n+1-i); B[i,j]
%o A342061     + (i+j)*if(i%2, if(j%2, C(i,j), D(j,i)), if(j%2, D(i,j)))
%o A342061     + sumdiv(i+j, d, if(d>1, eulerphi(d)*( if(i%d==0, E(i/d, j/d) ) + if(i%d==1, F((i-1)/d, (j+1)/d)) + if(j%d==1, F((j-1)/d, (i+1)/d)) )))
%o A342061    )/(2*n+2));
%o A342061 }
%o A342061 { my(A=T(10)); for(n=1, #A, print(A[n])) }
%Y A342061 Column k=3 is A001399(n-3).
%Y A342061 Row sums are A006402.
%Y A342061 Cf. A082680 (rooted), A239893, A342059.
%K A342061 nonn,tabl
%O A342061 2,8
%A A342061 _Andrew Howroyd_, Mar 30 2021