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 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). 31

%I

%S 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,

%T 10,19,31,46,63,81,99,116,131,143,151,154,151,143,131,116,99,81,63,46,

%U 31,19,10,4,1,1,5,15,34,65,111,174,255,354,470,601,744,895

%N 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).

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

%H G. C. Greubel, <a href="/A162499/b162499.txt">Rows n=0..20 of triangle, flattened</a>

%H A. V. Yurkin, <a href="http://www.mce.biophys.msu.ru/eng/archive/abstracts/mce19/sect1138/doc150220/">On similarity of systems of geometrical and arithmetic triangles</a>, in Mathematics, Computing, Education Conference XIX, 2012.

%H A. V. Yurkin, <a href="http://arxiv.org/abs/1302.6287">New view on the diffraction discovered by Grimaldi and Gaussian beams</a>, arXiv:1302.6287 [physics.optics], 2013.

%e Triangle begins:

%e 1

%e 1, 1, 1

%e 1, 2, 3, 3, 3, 3, 2, 1,

%e 1, 3, 6, 9, 12, 15, 17, 18, 18, 17, 15, 12, 9, 6, 3, 1,

%e 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

%e 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,

%e ...

%t 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 *)

%Y Rows give A162500, ...

%K nonn,tabf,look

%O 0,6

%A _N. J. A. Sloane_, Dec 02 2009

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Last modified February 7 21:52 EST 2023. Contains 360131 sequences. (Running on oeis4.)