A287610 Irregular triangle read by rows: universal linear relationships among polynomial means.

1, -1, 5, -6, 1, 1, -2, 2, -1, 2, -2, -1, 1, 77, -120, 60, -20, 3, 1, -3, 5, -5, 3, -1, 62, -75, -25, 75, -60, 23, 669, -1260, 1050, -700, 315, -84, 10, 3, -12, 28, -42, 42, -28, 12, -3, 1056, -2100, 1960, -1470, 735, -196, 0, 15, 4609, -10080, 11760, -11760, 8820, -4704, 1680, -360, 35, 1, -5, 15

Offset

1,3

Comments

Irregular triangular array of coefficients of universal linear relationships among means of all (complex-valued) polynomials, beginning with degree = 3 at top of triangle. Let phi(D,d,r) denote the mean of a generic degree D polynomial's order d derivative averaged over the (D-r) roots of the order r derivative of the polynomial. The tabulated coefficients, c(-), satisfy Sum_{k=1..(D-d-1)} c(k)*phi(D,d,k) = 0, with d always equal to 0.

Results have been computed in all degrees up to D=40, observing: (1) in all even degrees beyond 2, there is a single such linear relationship; (2) in all odd degrees beyond 3, there is a 2-dimensional family of such linear relationships.

In each row of the triangle, the sum of all positive coefficients equals the sum of all negative coefficients.

Links

Table of n, a(n) for n=1..65.

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

Triangle begins:
   1,   -1;
   5,   -6,   1;
   1,   -2,   2,  -1;
   2,   -2,  -1,   1;
  77, -120,  60, -20,   3;
   1,   -3,   5,  -5,   3,  -1;
  62,  -75, -25,  75, -60,  23;
  ...
Example 1: For any polynomial of degree D=3, it holds that 1*phi(3,0,1) - 1*phi(3,0,2) = 0.
Example 2: For any polynomial of degree D=4, it holds that 5*phi(4,0,1) - 6*phi(4,0,2) + 1*phi(4,0,3) = 0.
Example 3: For any polynomial of degree D=6, it holds that 77*phi(6,0,1) - 120*phi(6,0,2) + 60*phi(6,0,3) - 20*phi(6,0,4) + 3*phi(6,0,5) = 0.

Crossrefs

Sequence in context: A047774 A243108 A355186 * A298171 A046614 A371048

Adjacent sequences: A287607 A287608 A287609 * A287611 A287612 A287613

Keyword

sign,tabf

Author

Gregory Gerard Wojnar, May 27 2017

Extensions

More terms from Gregory Gerard Wojnar, Sep 24 2017

Edited by N. J. A. Sloane, Sep 30 2017

Status

approved