A290761 - OEIS

A290761

Irregular triangle, read by rows, of coefficients of polynomials that are the "nonstandard" factor of polynomials yielding the columns (up to sign) of triangle A290053, beginning with column 3.

%I #14 Sep 03 2017 22:16:09

%S 3,5,-6,16,1,7,16,28,0,15,225,1265,3707,7120,4900,-6480,27648,3,83,

%T 961,6201,24708,60700,87968,85056,0,63,2457,41580,404866,2532971,

%U 10651177,30102338,56577724,72856616,36562176,-51101568,298598400,9,531,14010,219106

%N Irregular triangle, read by rows, of coefficients of polynomials that are the "nonstandard" factor of polynomials yielding the columns (up to sign) of triangle A290053, beginning with column 3.

%C The polynomials come in pairs: first of odd degree; second of even degree 1 greater, whose constant term is always zero. Observations: All coefficients are positive except for the linear coefficients of the first polynomial in each pair, which are always negative. From the first of one pair to the first of the next pair, the degree always grows by 4. The "standard" factors of polynomials yielding the columns of triangle A290053 (beginning with column 3) are always of the form (1/A053657(k+2))*(N + k + 2) in odd rows of this triangle A290761, and of the form (N/A053657(k+2))*(N + k + 3)^2 in even rows of this triangle, where k is the row number. See examples.

%H G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, <a href="http://arxiv.org/abs/1706.08381">Universal Peculiar Linear Mean Relationships in All Polynomials</a>, arXiv:1706.08381 [math.GM], 2017.

%e The first rows of the triangle are parsed as follows:

%e 3, 5, -6, 16;

%e 1, 7, 16, 28, 0;

%e 15, 225, 1265, 3707, 7120, 4900, -6480, 27648;

%e 3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0;

%e 63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400;

%e 9, 531, 14010, 219106, 2266137, 16325259, 83797380, 307998768, 802828704, 1433652560, 1651979520, 1239918336, 0.

%e The associated full polynomials giving the columns of triangle A290053 are then:

%e (1/24) * (N + 3) * (3*N^3 + 5*N^2 - 6*N + 16);

%e (N/48) * (N + 5)^2 * (1*N^3 + 7*N^2 + 16*N + 28);

%e (1/5760) * (N + 5) * (15*N^7 + 225*N^6 + 1265*N^5 + 3707*N^4 + 7120*N^3 + 4900*N^2 - 6480*N + 27648);

%e (N/11520) * (N + 7)^2 * (3*N^7 + 83*N^6 + 961*N^5 + 6201*N^4 + 24708*N^3 + 60700*N^2 + 87968*N + 85056); etc.

%Y The first column of this triangle is A290030; alternating entries of the first column give A260326. See also triangle A290053, whose columns are A000012-A000096, A290061-A290071, A290127-A290723, etc.

%K sign,tabf

%O 1,1

%A _Gregory Gerard Wojnar_, Aug 09 2017