A307027 Number of (undirected) paths in the complete bipartite graph K_{m,n} (triangle read by rows with m = 1..n and n = 1..).

1, 3, 12, 6, 33, 135, 10, 72, 438, 2224, 15, 135, 1140, 8850, 55725, 21, 228, 2511, 27480, 265665, 2006316, 28, 357, 4893, 70462, 962010, 11158203, 98309827, 36, 528, 8700, 156768, 2818740, 46176816, 624859788, 6291829440, 45, 747, 14418, 313434, 7054875, 152212365, 2909139912

Offset: 1

Author

Eric W. Weisstein, Mar 20 2019

Keywords

• nonn
• tabl
• more

Examples

			1;
3,12;
6,33,135;
10,72,438,2224;
15,135,1140,8850,55725;
21,228,2511,27480,265665,2006316;
28,357,4893,70462,962010,11158203,98309827;
36,528,8700,156768,2818740,46176816,624859788,6291829440;
45,747,14418,313434,7054875,152212365,2909139912,...;
		

Links

• Eric Weisstein's World of Mathematics, Complete Bipartite Graph
• Eric Weisstein's World of Mathematics, Graph Path

Crossrefs

Cf. A288035 (K_{n,n} path count).

Formula

a(1, n) = binomial(n + 1, 2).

a(2, n) = n*(n^2 + 2).

a(3, n) = 3/2*n*(-3 + 11*n - 6*n^2 + 2*n^3).

a(4, n) = 2*n*(70 - 152*n + 123*n^2 - 42*n^3 + 6*n^4).

a(n, n) = A288035(n).