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A185095 Rectangular array read by antidiagonals: row q has generating function 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), where q=1,2,....

%I A185095 #44 Nov 30 2014 02:52:38
%S A185095 1,2,1,3,3,1,4,5,7,1,5,7,13,18,1,6,9,19,38,47,1,7,11,25,58,117,123,1,
%T A185095 8,13,31,78,187,370,322,1,9,15,37,98,257,622,1186,843,1,10,17,43,118,
%U A185095 327,874,2110,3827,2207,1,11,19,49,138,397,1126,3034,7252,12389,5778,1
%N A185095 Rectangular array read by antidiagonals: row q has generating function 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), where q=1,2,....
%C A185095 Row indices q begin with 1, column indices n begin with 0.
%H A185095 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 A185095 Conjecture. The n-th entry in row q is given by R_q(n) = 2^(2*n)*(sum_{j=1,...,n+1} (cos(j*Pi/(2*q+1)))^(2*n)), q >= 1, n >= 0.
%F A185095 Conjecture. G.f. for column n is of the form G_n(x) = H_n(x)/(1-x)^2, where H_n(x) is a polynomial in x, n >= 0.
%F A185095 Conjecture. 2*A185095(q,n) = A198632(2*q,n), q >= 1, n >= 0. - _L. Edson Jeffery_, Nov 23 2013
%e A185095 Array begins as
%e A185095 1,  1,  1,  1,   1,    1, ...
%e A185095 2,  3,  7, 18,  47,  123, ...
%e A185095 3,  5, 13, 38, 117,  370, ...
%e A185095 4,  7, 19, 58, 187,  622, ...
%e A185095 5,  9, 25, 78, 257,  874, ...
%e A185095 6, 11, 31, 98, 327, 1126, ...
%e A185095 ...
%Y A185095 Conjecture. Transpose of array A186740.
%Y A185095 Conjecture. Rows 0,1,2 (up to an offset) are A000012, A005248, A198636 (proved, see the Barbero, et al., reference there).
%Y A185095 Conjecture. Columns 0,1,2,3,4 (up to an offset) are A000027, A005408, A016921, A114698, A114646.
%Y A185095 Cf. A209235.
%K A185095 nonn,tabl
%O A185095 0,2
%A A185095 _L. Edson Jeffery_, Jan 23 2012