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A378336 Triangle read by rows: T(n,k) is the number of n node connected sensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

%I A378336 #9 Nov 25 2024 16:34:37
%S A378336 1,0,1,1,0,1,1,2,1,0,0,2,5,5,6,5,0,0,2,8,13,20,21,26,24,0,0,0,10,28,
%T A378336 55,79,104,119,147,133,0,0,0,7,45,126,230,360,491,625,735,892,846,0,0,
%U A378336 0,0,44,227,561,1066,1682,2430,3241,4074,4830,5876,5661
%N A378336 Triangle read by rows: T(n,k) is the number of n node connected sensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.
%C A378336 See A378103 for illustration of initial terms. This sequence does not consider planar maps to be equivalent to their reflections.
%C A378336 The planar maps considered are without loops or isthmuses.
%C A378336 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 up to orientation preserving isomorphisms.
%C A378336 The number of edges is n + k - 1.
%H A378336 Andrew Howroyd, <a href="/A378336/b378336.txt">Table of n, a(n) for n = 3..2306</a> (rows 3..50)
%F A378336 T(n,k) = 0 for n > 2*k + 1.
%F A378336 T(n,2*n-5) = A002709(n-3).
%F A378336 T(n,2*n-6) = A002710(n-4) for n >= 4.
%F A378336 T(n,2*n-7) = A002711(n-5) for n >= 5.
%e A378336 Triangle begins:
%e A378336 n\k | 1  2  3   4   5    6    7    8    9   10   11   12   13
%e A378336 ----+----------------------------------------------------------
%e A378336   3 | 1;
%e A378336   4 | 0, 1, 1;
%e A378336   5 | 0, 1, 1,  2,  1;
%e A378336   6 | 0, 0, 2,  5,  5,   6,   5;
%e A378336   7 | 0, 0, 2,  8, 13,  20,  21,  26,  24;
%e A378336   8 | 0, 0, 0, 10, 28,  55,  79, 104, 119, 147, 133;
%e A378336   9 | 0, 0, 0,  7, 45, 126, 230, 360, 491, 625, 735, 892, 846;
%e A378336   ...
%o A378336 (PARI) my(A=A378336rows(10)); for(i=1, #A, print(A[i])) \\ See PARI link in A378340 for program code.
%Y A378336 Row sums are A378335.
%Y A378336 Column sums are A378337.
%Y A378336 Antidiagonal sums are A378338.
%Y A378336 The final 3 terms of each row are in A002709, A002710, A002711.
%Y A378336 Cf. A262586 (2-connected), A341923 (3-connected), A378103, (unsensed), A378340 (achiral).
%K A378336 nonn,tabf
%O A378336 3,8
%A A378336 _Andrew Howroyd_, Nov 23 2024