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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.

%I A220672 #13 Jan 22 2020 23:50:29
%S A220672 -14,6,5,-12,3,46,-95,16,75,-69,24,-3,106,-520,928,-607,-351,894,-651,
%T A220672 234,-42,3,186,-1600,5840,-11355,11005,-1110,-9615,11580,-6906,2433,
%U A220672 -513,60,-3,286,-3775,22360,-75595,153515,-177565,77115,84495,-171324,145302,-75831,26235,-6057,900,-78,3
%N 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.
%C A220672 The row lengths sequence for this irregular triangle is 3*n+1 = A016777(n).
%C A220672 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.
%C A220672   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.
%C A220672   [-14, -3, 8, 4] is row m=2 of A217475.
%F A220672 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.
%e A220672 The array a(n,p) begins:
%e A220672 n\p   0     1    2     3     4    5     6    7    8  9
%e A220672 0:  -14
%e A220672 1:    6     5  -12     3
%e A220672 2:   46   -95   16    75   -69   24    -3
%e A220672 3:  106  -520  928  -607  -351  894  -651  234  -42  3
%e A220672 ...
%e A220672 Row n=4: [186, -1600, 5840, -11355, 11005, -1110, -9615, 11580, -6906, 2433, -513, 60, -3];
%e A220672 Row n=5: [286, -3775, 22360, -75595, 153515, -177565, 77115, 84495, -171324, 145302, -75831, 26235, -6057, 900, -78, 3].
%e A220672 Thus the conjecture is true at least for n=1..5.
%Y A220672 Cf. A220670, A220671.
%K A220672 sign,tabf
%O A220672 0,1
%A A220672 _Wolfdieter Lang_, Jan 14 2013