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A298368
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Triangle read by rows: T(n, k) = floor((n-1)/2)*floor(n/2)*floor((k-1)/2)*floor(k/2).
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1
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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
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refs;
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history;
text;
internal format)
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OFFSET
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1,9
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COMMENTS
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T(n, k) is conjectured by Zarankiewicz's conjecture to be the crossing number of the complete bipartite graph K_{k,n}.
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LINKS
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FORMULA
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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)
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EXAMPLE
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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;
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MAPLE
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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
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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