|
| |
|
|
A220672
|
|
Coefficients of powers of x^2 of polynomials, called h(2,n,x^2), appearing in a conjecture on alternating sums of fifth powers of odd-indexed Chebyshev S polynomials stated in A220671.
|
|
1
|
|
|
|
-14, 6, 5, -12, 3, 46, -95, 16, 75, -69, 24, -3, 106, -520, 928, -607, -351, 894, -651, 234, -42, 3, 186, -1600, 5840, -11355, 11005, -1110, -9615, 11580, -6906, 2433, -513, 60, -3, 286, -3775, 22360, -75595, 153515, -177565, 77115, 84495, -171324, 145302, -75831, 26235, -6057, 900, -78, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
COMMENTS
|
The row lengths sequence for this irregular triangle is 3*n+1 = A016777(n).
A generalized Melham conjecture involving fifth powers (m=2) of odd-indexed Chebyshev S polynomials (see A049310) is H(2,n,x^2):= (x^2-3)*(x^4-5*x^2+5)*sum(((-1)^k)*(S(2*k-1,x)/x)^(2*m+1), k=0..n)/((1 - (-1)^n*S(2*n,x))/x^2)^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), and h an integer polynomial of degree 3*n.
The present array a(n,p) appears as h(2,n,x^2) = sum(a(n,p)*x^(2*p),p=0..3*n), n >= 1. The entry a(0,0) := -14 has been used because, in accordance with the original Melham conjecture (see a comment on A220671), h(2,n,i^2), with the imaginary unit i, is conjectured to be -14, for all n >= 1.
[-14, -3, 8, 4] is row m=2 of A217475.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n,p) = [x^(2p)] h(0,2,n,x^2), with the polynomial h defined above in a comment. The conjecture is that h is an integer polynomial of degree 3n in x^2.
|
|
|
EXAMPLE
|
The array a(n,p) begins:
n\p 0 1 2 3 4 5 6 7 8 9
0: -14
1: 6 5 -12 3
2: 46 -95 16 75 -69 24 -3
3: 106 -520 928 -607 -351 894 -651 234 -42 3
...
Row n=4: [186, -1600, 5840, -11355, 11005, -1110, -9615, 11580, -6906, 2433, -513, 60, -3];
Row n=5: [286, -3775, 22360, -75595, 153515, -177565, 77115, 84495, -171324, 145302, -75831, 26235, -6057, 900, -78, 3].
Thus the conjecture is true at least for n=1..5.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
