0,6

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.

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

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) +...

(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)}

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

Sequence in context: A093869 A057431 A057060 this_sequence A064894 A003638 A094267

Adjacent sequences: A152802 A152803 A152804 this_sequence A152806 A152807 A152808

sign,tabf

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 28 2008