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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 (list; table; graph; refs; listen; history; text; internal format)
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 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.

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 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.

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

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Last modified December 5 12:51 EST 2020. Contains 338947 sequences. (Running on oeis4.)