OFFSET
0,4
COMMENTS
This sequence illustrates in part the identities:
* q-Cosh(x,q)^2 - q-Sinh(x,q)^2 = E_q(x,q) / e_q(x,q),
* q-cosh(x,q)^2 - q-sinh(x,q)^2 = e_q(x,q) / E_q(x,q).
Here the following q-analogs are employed (see MathWorld links):
q-cosh(x,q) = Sum_{n>=0} x^(2*n)/faq(2*n,q),
q-sinh(x,q) = Sum_{n>=0} x^(2*n+1)/faq(2*n+1,q),
and the dual expressions:
q-Cosh(x,q) = Sum_{n>=0} q^(n*(2*n-1))*x^(2*n)/faq(2*n,q),
q-Sinh(x,q) = Sum_{n>=0} q^(n*(2*n+1))*x^(2*n+1)/faq(2*n+1,q),
along with the dual q-exponential functions of x:
e_q(x,q) = Sum_{n>=0} x^n/faq(n,q),
E_q(x,q) = Sum_{n>=0} q^(n*(n-1)/2) * x^n/faq(n,q),
where
faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1000
Eric Weisstein, q-Exponential Function from MathWorld.
Eric Weisstein, q-Cosine Function from MathWorld.
Eric Weisstein, q-Sine Function from MathWorld.
Eric Weisstein, q-Factorial from MathWorld.
FORMULA
(1) G.f.: E_q(x,q) / e_q(x,q) at q=-x, where
e_q(x,-x) = Sum_{n>=0} x^n/Product(k=1, n, (1 - (-x)^k)/(1+x)),
E_q(x,-x) = Sum_{n>=0} (-x)^(n*(n-1)/2) * x^n/Product(k=1, n, (1 - (-x)^k)/(1+x)).
(2) G.f.: exp( -Sum_{n>=1} (1+x)^(2*n)/(1-x^(2*n)) * x^(2*n)/n ).
(3) G.f.: Product_{n>=1} (1 - x^(2*n)*(1+x)^2).
(4) Given g.f. A(x), A( (sqrt(5)-1)/2 ) = 0.
EXAMPLE
G.f.: A(x) = 1 - x^2 - 2*x^3 - 2*x^4 - 2*x^5 - x^6 + 2*x^7 + 5*x^8 +...
The g.f. may be expressed by:
(0) A(x) = q-Cosh(x,q)^2 - q-Sinh(x,q)^2 at q=-x, where
q-Cosh(x,-x) = 1 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8 - x^9 + 2*x^11 + 3*x^12 + 2*x^13 + x^14 + 2*x^15 +...
q-Sinh(x,-x) = x - x^6 - 2*x^7 - 2*x^8 - x^9 - x^12 - 2*x^13 - 2*x^14 + 4*x^16 +...
q-Cosh(x,-x)^2 = 1 - 2*x^3 - 2*x^4 - 2*x^5 - x^6 + x^8 + 2*x^9 + 5*x^10 + 10*x^11 + 13*x^12 +...
q-Sinh(x,-x)^2 = x^2 - 2*x^7 - 4*x^8 - 4*x^9 - 2*x^10 + x^12 + 2*x^13 + 4*x^14 + 6*x^15 +...
(1) A(x) = E_q(x,q) / e_q(x,q) at q=-x, where
e_q(x,-x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 11*x^6 + 17*x^7 + 28*x^8 + 48*x^9 + 80*x^10 + 128*x^11 + 204*x^12 +...
E_q(x,-x) = 1 + x - x^3 - x^4 - x^5 - 2*x^6 - 3*x^7 - 3*x^8 - 2*x^9 + 2*x^11 + 2*x^12 +...
(2) -log(A(x)) = (1+x)^2/(1-x^2)*x^2 + (1+x)^4/(1-x^4)*x^4/2 + (1+x)^6/(1-x^6)*x^6/3 + (1+x)^8/(1-x^8)*x^8/4 +...
(3) A(x) = (1 - x^2*(1+x)^2) * (1 - x^4*(1+x)^2) * (1 - x^6*(1+x)^2) * (1 - x^8*(1+x)^2) * (1 - x^10*(1+x)^2) *...
PROG
(PARI) /* (0) G.f. q-Cosh(x, q)^2 - q-Sinh(x, q)^2 at q=-x: */
{a(n)=local(Cosh_q=sum(k=0, n, (-x)^(k*(2*k-1))*x^(2*k)/(prod(j=1, 2*k, (1-(-x)^j)/(1+x))+x*O(x^n))), Sinh_q=sum(k=0, n, (-x)^(k*(2*k+1))*x^(2*k+1)/(prod(j=1, 2*k+1, (1-(-x)^j)/(1+x))+x*O(x^n)))); polcoeff(Cosh_q^2-Sinh_q^2, n)}
(PARI) /* (1) G.f. E_q(x, q) / e_q(x, q) at q=-x: */
{a(n)=local(e_q=sum(k=0, n, x^k/(prod(j=1, k, (1-(-x)^j)/(1+x))+x*O(x^n))), E_q=sum(k=0, n, (-x)^(k*(k-1)/2)*x^k/(prod(j=1, k, (1-(-x)^j)/(1+x))+x*O(x^n)))); polcoeff(E_q/e_q, n)}
(PARI) /* (1) G.f. E_q(x, q) / e_q(x, q) at q=-x: */
{a(n)=local(e_q=exp(sum(k=1, n, x^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n)), E_q=exp(sum(k=1, n, -(-x)^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n))); polcoeff(E_q/e_q, n)}
(PARI) /* (2) G.f. exp( -Sum_{n>=1} (1+x)^(2*n)/(1-x^(2*n)) * x^(2*n)/n): */
{a(n)=polcoeff( exp( -sum(m=1, n\2+1, (1+x)^(2*m)/(1-x^(2*m)+x*O(x^n))*x^(2*m)/m)), n)}
(PARI) /* (3) G.f. Product_{n>=1} (1 - x^(2*n)*(1+x)^2): */
{a(n)=polcoeff(prod(k=1, n, 1-(1+x)^2*x^(2*k)+x*O(x^n)), n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Oct 22 2011
STATUS
approved