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%I #4 Aug 25 2015 09:26:36
%S 1,-1,-1,1,-1,2,2,-1,-1,3,3,0,-3,-3,1,-1,4,3,-3,-4,-14,-4,-3,3,4,-1,
%T -1,5,2,-9,-11,-19,-16,-11,11,16,19,11,9,-2,-5,1,-1,6,0,-17,-18,-25,1,
%U -7,41,73,83,83,73,41,-7,1,-25,-18,-17,0,6,-1,-1,7,-3,-26,-20,-17,38,67
%N Expansion of 2/(1 + e_q(2x,q)) where e_q(2x,q) is the q-exponential of 2x, as a triangle of coefficients read by rows.
%C May be considered as coefficients of a q-analog of the tangent numbers (A000182).
%C The q-exponential of x is e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) where faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.
%F Sum_{k=0..n(n-1)/2} T(n,k) * exp(2*Pi*I*k/n) = -2^(n-1) for n>0.
%e Row n lists coefficients of powers of q ranging from q^0 to q^(n(n-1)/2).
%e Triangle begins:
%e 1;
%e -1;
%e -1,1;
%e -1,2,2,-1;
%e -1,3,3,0,-3,-3,1;
%e -1,4,3,-3,-4,-14,-4,-3,3,4,-1;
%e -1,5,2,-9,-11,-19,-16,-11,11,16,19,11,9,-2,-5,1;
%e -1,6,0,-17,-18,-25,1,-7,41,73,83,83,73,41,-7,1,-25,-18,-17,0,6,-1;
%e -1,7,-3,-26,-20,-17,38,67,115,184,223,217,198,84,0,-84,-198,-217,-223,-184,-115,-67,-38,17,20,26,3,-7,1;
%e -1,8,-7,-35,-13,12,110,161,258,271,261,219,33,-257,-638,-876,-1269,-1423,-1564,-1423,-1269,-876,-638,-257,33,219,261,271,258,161,110,12,-13,-35,-7,8,-1;
%e ...
%e EXPLICIT EXPANSION OF G.F.:
%e 1 - x + x^2*(-1 + q)/faq(2,q) + x^3*(-1 + 2*q + 2*q^2 - q^3)/faq(3,q) +
%e x^4*(-1 + 3*q + 3*q^2 - 3*q^4 - 3*q^5 + q^6)/faq(4,q) +
%e x^5*(-1 + 4*q + 3*q^2 - 3*q^3 - 4*q^4 - 14*q^5 - 4*q^6 - 3*q^7 + 3*q^8 + 4*q^9 - q^10)/faq(5,q) +...
%o (PARI) {T(n,k)=polcoeff(polcoeff(2/(1+sum(m=0,n,(2*x)^m/prod(j=1,m,(q^j-1)/(q-1))+x*O(x^(n+2)))),n,x)*prod(j=1,n,(q^j-1)/(q-1)),k,q)}
%o for(n=0,8,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))
%Y Cf. A000182 (row sums=signed tangent numbers); A152806 (unsigned row sums); A152807; A152800.
%K sign,tabf
%O 0,6
%A _Paul D. Hanna_, Dec 28 2008
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