A191928 Array read by antidiagonals: T(m,n) = floor(m/2)*floor((m-1)/2)*floor(n/2)*floor((n-1)/2).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 4, 4, 4, 0, 0, 0, 0, 0, 0, 6, 8, 8, 6, 0, 0, 0, 0, 0, 0, 9, 12, 16, 12, 9, 0, 0, 0, 0, 0, 0, 12, 18, 24, 24, 18, 12, 0, 0, 0, 0, 0, 0, 16, 24, 36, 36, 36, 24, 16, 0, 0, 0, 0, 0, 0, 20, 32, 48, 54, 54, 48, 32, 20, 0, 0, 0, 0, 0, 0, 25, 40, 64, 72, 81, 72, 64, 40, 25, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane, Jun 19 2011

Keywords

• nonn
• tabl

Comments

T(m,n) is conjectured to be the crossing number of the complete bipartite graph K_{m,n}.

Examples

			Array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 1, 2, 4, 6, 9, 12, ...
0, 0, 0, 2, 4, 8, 12, 18, 24, ...
0, 0, 0, 4, 8, 16, 24, 36, 48, ...
0, 0, 0, 6, 12, 24, 36, 54, 72, ...
0, 0, 0, 9, 18, 36, 54, 81, 108, ...
0, 0, 0, 12, 24, 48, 72, 108, 144, ...
		

Links

• D. McQuillan and R. B. Richter, A parity theorem for drawings of complete and bipartite graphs, Amer. Math. Monthly, 117 (2010), 267-273.

Crossrefs

Cf. A000241, A002620.

Programs

Maple

K:=(m,n)->floor(m/2)*floor((m-1)/2)*floor(n/2)*floor((n-1)/2);

PARI

T(n,k) = ((n-1)^2\4)*((k-1)^2\4);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print()); \\ Michel Marcus, Sep 30 2017

Formula

T(m,n) = A002620(m-1)*A002620(n-1). - Michel Marcus, Sep 30 2017