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A293107
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Irregular triangle read by rows: universal linear relationships among polynomial means for even degrees.
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1
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5, -6, 1, 77, -120, 60, -20, 3, 669, -1260, 1050, -700, 315, -84, 10, 4609, -10080, 11760, -11760, 8820, -4704, 1680, -360, 35, 55991, -13860, 207900, -277200, 291060, -232848, 138600, -59400, 17325, -3080, 252, 785633, -2162160, 3963960, -6606600, 8918910
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OFFSET
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1,1
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COMMENTS
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Irregular triangular array of coefficients of universal linear relationships among means of all even-degree (complex-valued) polynomials, beginning with degree = 4 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. The first row of the triangle has 3 entries, while each subsequent row has an additional 2 entries.
Results have been computed in all degrees up to D=49, 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.
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LINKS
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EXAMPLE
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Triangle begins:
5, -6, 1;
77, -120, 60, -20, 3;
669, -1260, 1050, -700, 315, -84, 10;
4609, -10080, 11760, -11760, 8820, -4704, 1680, -360, 35;
...
Example 1: 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 2: 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.
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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