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COMMENT
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In 1998, with help of planar polynomial differential systems, I discovered three bidimensional polynomials. Two of them are odd, the third, based on a degenerate case, is normal. In one dimension, it can be written D(n,z) = sum[((n+2)^2-(i+1)^2)*z/(i+1)] i=0..n Its roots are all complex when n is even and all except one if n is odd. They are equally distributed (thanks to Jean-Charles Faugare [Faugere?] in 2000). We write the first 4 lines and the transformed terms after multiplication by n!
.........3........................................3
.........8..5/2..................................16.....5
........15..12/2...7/3...........................90....36...14
........24..21/2..16/3..9/4.....................576...252..128..54.....
The numerators of the columns of the first array are well known (A005563, A028347, A028360). But the first vertical sequence and the one of the highest diagonal of the second array are unknown.
Read Faugere. Vertical numerators are n(n+2p)=(n+p)^2-p^2 (positive n,p) from A005563,A028347,A028560,A028566,A098603,A098847,A098848,A098849,A098850,A120071,A132764,A132766,A132768,A132770,A132772. Rows numerators are triangle A120070, for frequencies of the spectral lines of hydrogen atom: Lyman,A005563, Balmer,A061037, Paschen,A061039, Brackett,A061041, Pfund,A061043, Humphreys,A061045, Hansen-Strong,A061047, A061049, .. (numerators).See,in A120070, First ten rows and more. [From Paul Curtz (bpcrtz(AT)free.fr), Mar 14 2009]
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