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A290053
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Triangle read by rows: Polynomial coefficients per comment.
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6
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1, 1, 0, 1, -2, 3, 1, -5, 10, 0, 1, -9, 31, -39, 40, 1, -14, 77, -196, 252, 0, 1, -20, 162, -664, 1457, -1476, 1260, 1, -27, 303, -1809, 6168, -11772, 12176, 0, 1, -35, 520, -4250, 20773, -61595, 107730, -95400, 72576, 1, -44, 836, -8954, 59279, -249986
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OFFSET
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1,5
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COMMENTS
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Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rho-th derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f. [See arXiv:1706.08381 [math,GM], 2017.] The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=D-rho. These polynomials are of the form h_D(N) = ((-1)^D/(D-1)!)(D-N)N^chi*g_D(N) where chi = (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D-2-chi). Then a(n) are the coefficients of the polynomials N^chi*g_D(N), starting at D=2. The leading term of each row is 1 (polynomials are monic). The final terms in all even rows are 0. In each row, terms alternate in sign.
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LINKS
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Gregory Gerard Wojnar, java_program which (1) creates Maple program to create polynomial referenced in Comment, and (2) creates list of polynomial portion's coefficients (without trailing 0 constant term is odd degree cases) which constitute the rows of this triangle. Each run of the program is for a single degree; to change the degree the user must modify the value of "level" in line 393 of the java code.
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EXAMPLE
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Triangle begins:
1;
1, 0;
1, -2, 3;
1, -5, 10, 0;
1, -9, 31, -39, 40;
1, -14, 77, -196, 252, 0;
1, -20, 162, -664, 1457, -1476, 1260;
1, -27, 303, -1809, 6168, -11772, 12176, 0;
1, -35, 520, -4250, 20773, -61595, 107730, -95400, 72576;
...
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CROSSREFS
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The final terms in odd-numbered rows are A110468.
The negation of the second column give A000096.
The 3rd column is A290061; negation of 4th column is A290071; 5th column is A290127. Up to sign, all columns are given by polynomials described in the comments and examples of triangle A290761.
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KEYWORD
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AUTHOR
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STATUS
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approved
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