This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A290053 #24 Sep 03 2017 22:15:12 %S A290053 1,1,0,1,-2,3,1,-5,10,0,1,-9,31,-39,40,1,-14,77,-196,252,0,1,-20,162, %T A290053 -664,1457,-1476,1260,1,-27,303,-1809,6168,-11772,12176,0,1,-35,520, %U A290053 -4250,20773,-61595,107730,-95400,72576,1,-44,836,-8954,59279,-249986 %N A290053 Triangle read by rows: Polynomial coefficients per comment. %C A290053 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. %H A290053 G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, <a href="http://arxiv.org/abs/1706.08381">Universal Peculiar Linear Mean Relationships in All Polynomials</a>, arXiv:1706.08381 [math.GM], 2017. %H A290053 Gregory Gerard Wojnar, <a href="/A290053/a290053.java.txt">java_program</a> 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. %e A290053 Triangle begins: %e A290053 1; %e A290053 1, 0; %e A290053 1, -2, 3; %e A290053 1, -5, 10, 0; %e A290053 1, -9, 31, -39, 40; %e A290053 1, -14, 77, -196, 252, 0; %e A290053 1, -20, 162, -664, 1457, -1476, 1260; %e A290053 1, -27, 303, -1809, 6168, -11772, 12176, 0; %e A290053 1, -35, 520, -4250, 20773, -61595, 107730, -95400, 72576; %e A290053 ... %Y A290053 The final terms in odd-numbered rows are A110468. %Y A290053 The negation of the second column give A000096. %Y A290053 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. %K A290053 sign,tabl %O A290053 1,5 %A A290053 _Gregory Gerard Wojnar_, Jul 19 2017