login
Triangle read by rows: T(n,k) = number of nonisomorphic unlabeled planar graphs with n >= 1 nodes and 0 <= k <= 3n-6 edges.
5

%I #17 Sep 04 2023 11:34:56

%S 1,1,1,1,1,1,1,1,1,2,3,2,1,1,1,1,2,4,6,6,6,4,2,1,1,1,2,5,9,15,21,24,

%T 24,20,13,5,2,1,1,2,5,10,21,41,65,97,130,144,135,98,51,16,5,1,1,2,5,

%U 11,24,56,115,221,401,658,956,1217,1264,1042,631,275,72,14,1,1,2,5

%N Triangle read by rows: T(n,k) = number of nonisomorphic unlabeled planar graphs with n >= 1 nodes and 0 <= k <= 3n-6 edges.

%C Planar graphs with n >= 3 nodes have at most 3n-6 edges. - _Charles R Greathouse IV_, Feb 18 2013

%D R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

%D R. J. Wilson, Introduction to Graph Theory. Academic Press, NY, 1972, p. 162.

%H F. Harary, <a href="http://dx.doi.org/10.1090/S0002-9947-1955-0068198-2">The number of linear, directed, rooted, and connected graphs</a>, Trans. Amer. Math. Soc. 78 (1955), 445-463. (MR0068198) See page 457, equation (2.9).

%F From _Michael Somos_, Aug 23 2015: (Start)

%F Sum_{k} T(n, k) = A005470(n) if n >= 1.

%F log(1 + A(x, y)) = Sum_{n>0} B(x^n, y^n) / n where A(x, y) = Sum_{n>0, k>=0} T(n,k) * x^n * y^k and similarly B(x, y) with A049334. (End)

%e Triangle starts

%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 12

%e --:-- -- -- -- -- -- -- -- -- -- -- -- --

%e 1: 1

%e 2: 1 1

%e 3: 1 1 1 1

%e 4: 1 1 2 3 2 1 1

%e 5: 1 1 2 4 6 6 6 4 2 1

%e 6: 1 1 2 5 9 15 21 24 24 20 13 5 2

%Y Cf. A005470 (row sums), A008406, A049334.

%K nonn,tabf,nice

%O 1,10

%A _Brendan McKay_