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A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2*n)*(Sum_{j=1..n+1} (cos(j*Pi/(2*q+1)))^(2*n)), n >= 0, q >= 1.

%I A186740 #42 Apr 21 2017 04:01:22
%S A186740 1,1,2,1,3,3,1,7,5,4,1,18,13,7,5,1,47,38,19,9,6,1,123,117,58,25,11,7,
%T A186740 1,322,370,187,78,31,13,8,1,843,1186,622,257,98,37,15,9,1,2207,3827,
%U A186740 2110,874,327,118,43,17,10,1,5778,12389,7252,3034,1126,397,138,49,19,11
%N A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2*n)*(Sum_{j=1..n+1} (cos(j*Pi/(2*q+1)))^(2*n)), n >= 0, q >= 1.
%C A186740 Row indices n begin with 0, column indices q begin with 1.
%H A186740 Andrew Howroyd, <a href="/A186740/b186740.txt">Table of n, a(n) for n = 0..1275</a>
%H A186740 S. Barbero, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barbero/barbero15.html">Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery</a>, Journal of Integer Sequences, 17 (2014), #14.3.8.
%F A186740 Conjecture: G.f. for column q is F_q(x) = (Sum_{r=0..q-1} ((q-r)*(-1)^r*binomial(2*q-r,r)*x^r)) / (Sum_{s=0..q} ((-1)^s*binomial(2*q-s,s)*x^s)), q >= 1.
%F A186740 Conjecture: G.f. for n-th row is of the form G_n(x) = H_n(x)/(1-x)^2, where H_n(x) is a polynomial in x.
%e A186740 Array begins:
%e A186740 1    2     3     4     5     6     7     8     9 ...
%e A186740 1    3     5     7     9    11    13    15    17 ...
%e A186740 1    7    13    19    25    31    37    43    49 ...
%e A186740 1   18    38    58    78    98   118   138   158 ...
%e A186740 1   47   117   187   257   327   397   467   537 ...
%e A186740 1  123   370   622   874  1126  1378  1630  1882 ...
%e A186740 1  322  1186  2110  3034  3958  4882  5806  6730 ...
%e A186740 1  843  3827  7252 10684 14116 17548 20980 24412 ...
%e A186740 1 2207 12389 25147 38017 50887 63757 76627 89497 ...
%e A186740 ...
%e A186740 As a triangle:
%e A186740 1,
%e A186740 1,  2,
%e A186740 1,  3,  3,
%e A186740 1,  7,  5,  4,
%e A186740 1, 18, 13,  7, 5,
%e A186740 1, 47, 38, 19, 9, 6,
%e A186740 ...
%Y A186740 Conjecture: Transpose of array A185095.
%Y A186740 Conjecture: Columns 0,1,2 (up to an offset) are A000012, A005248, A198636 (proved, see the Barbero, et al., reference there).
%Y A186740 Conjecture: Rows 0,1,2,3,4 (up to an offset) are A000027, A005408, A016921, A114698, A114646.
%Y A186740 Cf. A209235.
%K A186740 nonn,tabl
%O A186740 0,3
%A A186740 _L. Edson Jeffery_, Jan 21 2012