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A378103 Triangle read by rows: T(n,k) is the number of n-node connected unsensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

%I A378103 #15 Nov 25 2024 20:44:47
%S A378103 1,0,1,1,0,1,1,2,1,0,0,2,4,4,5,4,0,0,2,6,10,14,14,18,16,0,0,0,7,18,35,
%T A378103 49,63,69,88,78,0,0,0,5,28,74,131,204,274,345,396,489,457,0,0,0,0,26,
%U A378103 126,304,574,893,1290,1708,2137,2503,3071,2938,0,0,0,0,13,159,582,1396,2613,4274,6270,8709,11433,14227,16905,20667,20118
%N A378103 Triangle read by rows: T(n,k) is the number of n-node connected unsensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.
%C A378103 The planar maps considered are without loops or isthmuses.
%C A378103 In other words, a(n) is the number of embeddings in the plane of connected bridgeless planar simple graphs with n vertices and k triangular internal faces.
%C A378103 The number of edges is n + k - 1.
%C A378103 The nonzero terms in row n range from k = floor(n/2) through 2*n-5 and, thus, the number of nonzero terms is 2n - floor(n/2) - 4 = A001651(n-2).
%H A378103 Andrew Howroyd, <a href="/A378103/b378103.txt">Table of n, a(n) for n = 3..2306</a> (rows 3..50)
%H A378103 Ya-Ping Lu, <a href="/A378103/a378103.pdf">Illustration of initial terms</a>
%F A378103 T(n, 2*n-5) = A002713(n-3).
%F A378103 T(n,k) = (A378336(n,k) + A378340(n,k))/2.
%e A378103 Triangle begins:
%e A378103 n\k        1     2     3     4     5     6     7     8     9    10    11
%e A378103 ----     ----  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
%e A378103 3          1
%e A378103 4          0     1     1
%e A378103 5          0     1     1     2     1
%e A378103 6          0     0     2     4     4     5     4
%e A378103 7          0     0     2     6    10    14    14    18    16
%e A378103 8          0     0     0     7    18    35    49    63    69    88    78
%o A378103 (PARI) my(A=A378103rows(10)); for(i=1, #A, print(A[i])) \\ See PARI link in A378340 for program code. - _Andrew Howroyd_, Nov 25 2024
%Y A378103 Row sums are A377785.
%Y A378103 Cf. A001651, A002713, A003094, A169808, A378336 (sensed), A378340 (achiral).
%Y A378103 The final 3 terms of each row are in A002713, A005500, A005501.
%K A378103 nonn,tabf
%O A378103 3,8
%A A378103 _Ya-Ping Lu_, Nov 16 2024
%E A378103 a(39) onwards from _Andrew Howroyd_, Nov 25 2024