A162499 Triangle read by rows in which row n gives coefficients of the expansion of the polynomial Product( (1-x^(3*k))/(1-x), k=1..n).

1, 1, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 3, 6, 9, 12, 15, 17, 18, 18, 17, 15, 12, 9, 6, 3, 1, 1, 4, 10, 19, 31, 46, 63, 81, 99, 116, 131, 143, 151, 154, 151, 143, 131, 116, 99, 81, 63, 46, 31, 19, 10, 4, 1, 1, 5, 15, 34, 65, 111, 174, 255, 354, 470, 601, 744, 895

Offset: 0

Author

N. J. A. Sloane, Dec 02 2009

Keywords

• nonn
• tabf
• look

Examples

			Triangle begins:
1
1, 1, 1
1, 2, 3, 3, 3, 3, 2, 1,
1, 3, 6, 9, 12, 15, 17, 18, 18, 17, 15, 12, 9, 6, 3, 1,
1, 4, 10, 19, 31, 46, 63, 81, 99, 116, 131, 143, 151, 154, 151, 143, 131, 116, 99, 81, 63, 46, 31, 19, 10, 4, 1
1, 5, 15, 34, 65, 111, 174, 255, 354, 470, 601, 744, 895, 1049, 1200, 1342, 1469, 1575, 1655, 1705, 1722, 1705, 1655, 1575, 1469, 1342, 1200, 1049, 895, 744, 601, 470, 354, 255, 174, 111, 65, 34, 15, 5, 1,
...
		

References

• A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.

Links

• G. C. Greubel, Rows n=0..20 of triangle, flattened
• A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
• A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv:1302.6287 [physics.optics], 2013.

Crossrefs

Rows give A162500, ...

Programs

Mathematica

row[n_] := CoefficientList[Product[(1 - x^(3*k))/(1 - x), {k, 1, n}], x]; Table[row[n], {n, 0, 5}] // Flatten (* Jean-François Alcover, Sep 19 2016 *)