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 A220671 #26 Feb 21 2025 06:19:08 %S A220671 -14,15,-20,8,-1,55,-170,221,-153,59,-12,1,115,-670,1773,-2696,2549, %T A220671 -1538,589,-138,18,-1,195,-1850,8215,-21530,36330,-41110,31865,-17080, %U A220671 6314,-1579,255,-24,1,295,-4150,27735,-110795,289540,-518290,654595,-595805,396316,-193906,69641,-18129,3327,-408,30,-1 %N A220671 Coefficient array for powers of x^2 of polynomials appearing in a generalized Melham conjecture on alternating sums of fifth powers of Chebyshev S polynomials with odd indices. %C A220671 The row lengths sequence is 3*n + 1 = A016777(n). %C A220671 For the generalized Melham conjecture and links to the references concerned with the Melham conjecture on sums of fifth powers of even-indexed Fibonacci numbers see a comment under A220670. %C A220671 Here the conjecture is considered for m=2 (fifth powers): H(2,n,x^2):= product(tau(j,x), j=0..2) * sum(((-1)^k)*(S(2*k-1,x)/x)^5, k=0..n) / (P(n,x^2)^2), with P(n,x^2):= (1 - (-1)^n*S(2*n,x))/x^2. For tau(j,x):= 2*T(2*j+1,x/2)/x, with Chebyshev's T polynomials see a Oct 23 2012 comment on A111125. For the polynomials P see signed A109954. The conjecture is that H(2,n,x^2) is an integer polynomial of degree 3*n: H(2,n,x^2) = sum(a(n,p)*x^(2*p), p=0..3*n), n >= 1. %C A220671 If one puts x = i (the imaginary unit) one obtains the original Melham conjecture for Fibonacci numbers F = A000045. %C A220671 H(2,n,-1) = +44*sum(F(2*k)^5,k=0..n)/(1+F(2*n+1))^2, n>=1, which is conjectured to be -14 - 3*y(n) + 8*y(n)^2 + 4*y(n)^3, with y(n):=F(2*n+1) (see row m=2 of A217475). %C A220671 It is conjectured that H(2,n,x^2) = h(2,n,x^2) - 3*z(n) + 8*z(n)^2 + 4*z(n)^3, with z(n):= ((-1)^n)*S(2*n,x), with h an integer polynomial of degree 3*n. See A220672 for the coefficients of h(2,n,x^2) for n = 1..5. Because h(2,n,-1) = -14 by the usual Melham conjecture, we put h(2,0,x^2) = -14. %F A220671 a(n,p) = [x^(2*p)] H(2,n,x^2), n>=1, with H(2,n,x^2) defined in a comment above. a(0,0) has been put to -14 ad hoc. %e A220671 The array a(n,p) begins: %e A220671 n\p 0 1 2 3 4 5 6 7 8 9 10 11 12 %e A220671 0: -14 %e A220671 1: 15 -20 8 -1 %e A220671 2: 55 -170 221 -153 59 -12 1 %e A220671 3: 115 -670 1773 -2696 2549 -1538 589 -138 18 -1 %e A220671 4: 195 -1850 8215 -21530 36330 -41110 31865 -17080 6314 -1579 255 -24 1 %e A220671 ... %e A220671 Row n=5: [295, -4150, 27735, -110795, 289540, -518290, 654595, -595805, 396316, -193906, 69641, -18129, 3327, -408, 30, -1], %e A220671 Row n=6: [415, -8120, 76118, -429531, 1599441, -4125672, 7621983, -10350335, 10539787, -8164410, 4853792, -2222153, 781514, -209172, 41823, -6047, 597, -36, 1]. %Y A220671 Cf. A220670, A217475, A220672. %K A220671 sign,tabf %O A220671 1,1 %A A220671 _Wolfdieter Lang_, Jan 11 2013