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A220671
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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.
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2
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-14, 15, -20, 8, -1, 55, -170, 221, -153, 59, -12, 1, 115, -670, 1773, -2696, 2549, -1538, 589, -138, 18, -1, 195, -1850, 8215, -21530, 36330, -41110, 31865, -17080, 6314, -1579, 255, -24, 1, 295, -4150, 27735, -110795, 289540, -518290, 654595, -595805, 396316, -193906, 69641, -18129, 3327, -408, 30, -1
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OFFSET
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1,1
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COMMENTS
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The row lengths sequence is 3*n + 1 = A016777(n).
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.
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.
If one puts x = i (the imaginary unit) one obtains the original Melham conjecture for Fibonacci numbers F = A000045.
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).
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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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The array a(n,p) begins:
n\p 0 1 2 3 4 4 5 6 7 8 9
0: -14
1: 15 -20 8 -1
2: 55 -170 221 -153 59 -12 1
3: 115 -670 1773 -2696 2549 -1538 589 -138 18 -1
...
Row n=4: 195, -1850, 8215, -21530, 36330, -41110, 31865, -17080, 6314, -1579, 255, -24, 1],
Row n=5: [295, -4150, 27735, -110795, 289540, -518290, 654595, -595805, 396316, -193906, 69641, -18129, 3327, -408, 30, -1],
Row n=6: [415, -8120, 76118, -429531, 1599441, -4125672, 7621983, -10350335, 10539787, -8164410, 4853792, -2222153, 781514, -209172, 41823, -6047, 597, -36, 1].
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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