A298368 Triangle read by rows: T(n, k) = floor((n-1)/2)*floor(n/2)*floor((k-1)/2)*floor(k/2).

0, 0, 0, 0, 0, 1, 0, 0, 2, 4, 0, 0, 4, 8, 16, 0, 0, 6, 12, 24, 36, 0, 0, 9, 18, 36, 54, 81, 0, 0, 12, 24, 48, 72, 108, 144, 0, 0, 16, 32, 64, 96, 144, 192, 256, 0, 0, 20, 40, 80, 120, 180, 240, 320, 400, 0, 0, 25, 50, 100, 150, 225, 300, 400, 500, 625

Offset: 1

Author

Eric W. Weisstein, Jan 17 2018

Keywords

• nonn
• easy
• tabl
• nice

Comments

T(n, k) is conjectured by Zarankiewicz's conjecture to be the crossing number of the complete bipartite graph K_{k,n}.

Examples

			First rows are given by:
  0;
  0,   0;
  0,   0,   1;
  0,   0,   2,   4;
  0,   0,   4,   8,  16;
  0,   0,   6,  12,  24,  36;
  0,   0,   9,  18,  36,  54,  81;
  0,   0,  12,  24,  48,  72, 108, 144;
  0,   0,  16,  32,  64,  96, 144, 192, 256;
  0,   0,  20,  40,  80, 120, 180, 240, 320, 400;
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)
• Eric Weisstein's World of Mathematics, Graph Crossing Number.
• Eric Weisstein's World of Mathematics, Zarankiewicz's Conjecture.

Crossrefs

Cf. A030179, A191928.

Programs

Maple

seq(seq(floor((k-1)/2)*floor(k/2)*floor((n-1)/2)*floor(n/2),k=1..n),n=1..12); # Robert Israel, Jan 17 2018

Mathematica

Table[Floor[(m - 1)/2] Floor[m/2] Floor[(n - 1)/2] Floor[n/2], {n, 11}, {m, n}] // Flatten
Table[Times @@ Floor[{m, m - 1, n, n - 1}/2], {n, 11}, {m, n}] // Flatten

Formula

T(n,n) = A030179(n).

From Robert Israel, Jan 17 2018: (Start)

T(n,k) = A002620(n-1)*A002620(k-1).

G.f. as triangle: x^3*y^3*(1+2*x*y+6*x^2*y^2-4*x^3*y-8*x^3*y^2+2*x^4*y+2*x^3*y^3-4*x^4*y^2-2*x^4*y^3+4*x^5*y^2+ x^4*y^4-4*x^5*y^3-2*x^5*y^4+4*x^6*y^3+2*x^7*y^4)/

((1-x*y)^5*(1+x*y)^3*(1-x)^3*(1+x)). (End)