

A290053


Triangle read by rows: Polynomial coefficients per comment.


6



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

1,5


COMMENTS

Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rhoth 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:=Drho. These polynomials are of the form h_D(N) = ((1)^D/(D1)!)(DN)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 (D2chi). 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.


LINKS

Table of n, a(n) for n=1..51.
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.
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.


EXAMPLE

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;
...


CROSSREFS

The final terms in oddnumbered 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.
Sequence in context: A255689 A030103 A255589 * A105640 A090299 A060693
Adjacent sequences: A290050 A290051 A290052 * A290054 A290055 A290056


KEYWORD

sign,tabl


AUTHOR

Gregory Gerard Wojnar, Jul 19 2017


STATUS

approved



