A039735 Triangle read by rows: T(n,k) = number of nonisomorphic unlabeled planar graphs with n >= 1 nodes and 0 <= k <= 3n-6 edges.

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

Offset: 1

Author

Brendan McKay

Keywords

• nonn
• tabf
• nice

Comments

Planar graphs with n >= 3 nodes have at most 3n-6 edges. - Charles R Greathouse IV, Feb 18 2013

Examples

			Triangle starts
n\k 0  1  2  3  4  5  6  7  8  9 10 11 12
--:-- -- -- -- -- -- -- -- -- -- -- -- --
1:  1
2:  1  1
3:  1  1  1  1
4:  1  1  2  3  2  1  1
5:  1  1  2  4  6  6  6  4  2  1
6:  1  1  2  5  9 15 21 24 24 20 13  5  2
		

References

• R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
• R. J. Wilson, Introduction to Graph Theory. Academic Press, NY, 1972, p. 162.

Links

• F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Amer. Math. Soc. 78 (1955), 445-463. (MR0068198) See page 457, equation (2.9).

Crossrefs

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

Formula

From Michael Somos, Aug 23 2015: (Start)

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

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)