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

0,6

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.

Example

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,
...

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 *)

Crossrefs

Rows give A162500, ...

Sequence in context: A127661 A358617 A008968 * A350857 A135715 A089326

Adjacent sequences: A162496 A162497 A162498 * A162500 A162501 A162502

Keyword

nonn,tabf,look

Author

N. J. A. Sloane, Dec 02 2009

Status

approved