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
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
KEYWORD
sign,tabl
AUTHOR
Gregory Gerard Wojnar, Jul 19 2017
STATUS
approved