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.

3, 5, -6, 16, 1, 7, 16, 28, 0, 15, 225, 1265, 3707, 7120, 4900, -6480, 27648, 3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0, 63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400, 9, 531, 14010, 219106

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..42.

G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin,  Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

Example

The first rows of the triangle are parsed as follows:
3, 5, -6, 16;
1, 7, 16, 28, 0;
15, 225, 1265, 3707, 7120, 4900, -6480, 27648;
3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0;
63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400;
9, 531, 14010, 219106, 2266137, 16325259, 83797380, 307998768, 802828704, 1433652560, 1651979520, 1239918336, 0.
The associated full polynomials giving the columns of triangle A290053 are then:
(1/24) * (N + 3) * (3*N^3 + 5*N^2 - 6*N + 16);
(N/48) * (N + 5)^2 * (1*N^3 + 7*N^2 + 16*N + 28);
(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);
(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.

Crossrefs

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.

Sequence in context: A099678 A287687 A348658 * A192119 A050563 A282809

Adjacent sequences: A290758 A290759 A290760 * A290762 A290763 A290764

Keyword

sign,tabf

Author

Gregory Gerard Wojnar, Aug 09 2017

Status

approved