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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 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.)