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%I #37 Sep 30 2017 18:00:44
%S 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,
%T -75,-25,75,-60,23,669,-1260,1050,-700,315,-84,10,3,-12,28,-42,42,-28,
%U 12,-3,1056,-2100,1960,-1470,735,-196,0,15,4609,-10080,11760,-11760,8820,-4704,1680,-360,35,1,-5,15
%N Irregular triangle read by rows: universal linear relationships among polynomial means.
%C 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.
%C 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.
%C In each row of the triangle, the sum of all positive coefficients equals the sum of all negative coefficients.
%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 Triangle begins:
%e 1, -1;
%e 5, -6, 1;
%e 1, -2, 2, -1;
%e 2, -2, -1, 1;
%e 77, -120, 60, -20, 3;
%e 1, -3, 5, -5, 3, -1;
%e 62, -75, -25, 75, -60, 23;
%e ...
%e Example 1: For any polynomial of degree D=3, it holds that 1*phi(3,0,1) - 1*phi(3,0,2) = 0.
%e 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.
%e 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.
%K sign,tabf
%O 1,3
%A _Gregory Gerard Wojnar_, May 27 2017
%E More terms from _Gregory Gerard Wojnar_, Sep 24 2017
%E Edited by _N. J. A. Sloane_, Sep 30 2017
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