A152805 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.

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, -1, 5, 2, -9, -11, -19, -16, -11, 11, 16, 19, 11, 9, -2, -5, 1, -1, 6, 0, -17, -18, -25, 1, -7, 41, 73, 83, 83, 73, 41, -7, 1, -25, -18, -17, 0, 6, -1, -1, 7, -3, -26, -20, -17, 38, 67

Offset: 0

Author

Paul D. Hanna, Dec 28 2008

Keywords

• sign
• tabf

Comments

May be considered as coefficients of a q-analog of the tangent numbers (A000182).

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.

Examples

			Row n lists coefficients of powers of q ranging from q^0 to q^(n(n-1)/2).
Triangle begins:
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;
-1,5,2,-9,-11,-19,-16,-11,11,16,19,11,9,-2,-5,1;
-1,6,0,-17,-18,-25,1,-7,41,73,83,83,73,41,-7,1,-25,-18,-17,0,6,-1;
-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;
-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;
...
EXPLICIT EXPANSION OF G.F.:
1 - x + x^2*(-1 + q)/faq(2,q) + x^3*(-1 + 2*q + 2*q^2 - q^3)/faq(3,q) +
x^4*(-1 + 3*q + 3*q^2 - 3*q^4 - 3*q^5 + q^6)/faq(4,q) +
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) +...
		

Crossrefs

Cf. A000182 (row sums=signed tangent numbers); A152806 (unsigned row sums); A152807; A152800.

Programs

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)}
for(n=0,8,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))

Formula

Sum_{k=0..n(n-1)/2} T(n,k) * exp(2*Pi*I*k/n) = -2^(n-1) for n>0.