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%I #20 Oct 22 2020 04:48:11
%S 1,0,1,2,3,6,10,16,28,48,79,130,215,356,587,960,1566,2558,4176,6804,
%T 11066,17978,29198,47406,76916,124716,202152,327600,530775,859734,
%U 1392265,2254336,3649840,5908632,9564377,15480706,25055322,40549980,65624224,106199306,171856555,278099872
%N G.f.: q-cosh(x,q)^2 - q-sinh(x,q)^2 at q=-x.
%C This sequence illustrates in part the identities:
%C * q-cosh(x,q)^2 - q-sinh(x,q)^2 = e_q(x,q) / E_q(x,q),
%C * q-Cosh(x,q)^2 - q-Sinh(x,q)^2 = E_q(x,q) / e_q(x,q).
%C Here the following q-analogs are employed (see MathWorld links):
%C q-cosh(x,q) = Sum_{n>=0} x^(2*n)/faq(2*n,q),
%C q-sinh(x,q) = Sum_{n>=0} x^(2*n+1)/faq(2*n+1,q),
%C and the dual expressions:
%C q-Cosh(x,q) = Sum_{n>=0} q^(n*(2*n-1))*x^(2*n)/faq(2*n,q),
%C q-Sinh(x,q) = Sum_{n>=0} q^(n*(2*n+1))*x^(2*n+1)/faq(2*n+1,q),
%C along with the dual q-exponential functions of x:
%C e_q(x,q) = Sum_{n>=0} x^n/faq(n,q),
%C E_q(x,q) = Sum_{n>=0} q^(n*(n-1)/2) * x^n/faq(n,q),
%C where
%C faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/q-ExponentialFunction.html">q-Exponential Function</a> from MathWorld.
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/q-Cosine.html">q-Cosine Function</a> from MathWorld.
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/q-Sine.html">q-Sine Function</a> from MathWorld.
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/q-Factorial.html">q-Factorial Function</a> from MathWorld.
%F (1) G.f.: e_q(x,q) / E_q(x,q) at q=-x, where
%F e_q(x,-x) = Sum_{n>=0} x^n/Product_{k=1..n} (1-(-x)^k)/(1+x)),
%F E_q(x,-x) = Sum_{n>=0} (-x)^(n*(n-1)/2) * x^n/Product_{k=1..n} (1-(-x)^k)/(1+x)).
%F (2) G.f.: exp( Sum_{n>=1} (1+x)^(2*n)/(1-x^(2*n)) * x^(2*n)/n ).
%F (3) G.f.: Product_{n>=1} 1/(1 - x^(2*n)*(1+x)^2).
%F (4) Limit a(n+1)/a(n) = phi = (sqrt(5)+1)/2 with Limit a(n)/phi^n = 0.75149846280232258786564518960536101986114488526276981847216113150440...
%F Limit a(n)/phi^n = phi / (2*sqrt(5)) * Product_{k>=2} 1/(1 - phi^(2 - 2*k)). - _Vaclav Kotesovec_, Oct 22 2020
%e G.f.: A(x) = 1 + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 16*x^7 + 28*x^8 +...
%e The g.f. may be expressed by:
%e (0) A(x) = q-cosh(x,q)^2 - q-sinh(x,q)^2 at q=-x, where
%e q-cosh(x,-x) = 1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 6*x^6 + 9*x^7 + 15*x^8 + 25*x^9 + 41*x^10 + 66*x^11 + 105*x^12 +...
%e q-sinh(x,-x) = x + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 8*x^7 + 13*x^8 + 23*x^9 + 39*x^10 + 62*x^11 + 99*x^12 +...
%e q-cosh(x,-x)^2 = 1 + 2*x^2 + 2*x^3 + 5*x^4 + 10*x^5 + 17*x^6 + 30*x^7 + 54*x^8 + 96*x^9 + 170*x^10 + 296*x^11 + 510*x^12 +...
%e q-sinh(x,-x)^2 = x^2 + 2*x^4 + 4*x^5 + 7*x^6 + 14*x^7 + 26*x^8 + 48*x^9 + 91*x^10 + 166*x^11 + 295*x^12 +...
%e (1) A(x) = e_q(x,q) / E_q(x,q) at q=-x, where
%e 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 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 +...
%e (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 +...
%e (3) A(x) = 1/((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) *...).
%t nmax = 50; CoefficientList[Series[Product[1/(1 - x^(2*k)*(1+x)^2), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 22 2020 *)
%o (PARI) /* (0) G.f. q-cosh(x,q)^2 - q-sinh(x,q)^2 at q=-x: */
%o {a(n)=local(cosh_q=sum(k=0, n, x^(2*k)/(prod(j=1, 2*k, (1-(-x)^j)/(1+x))+x*O(x^n))),sinh_q=sum(k=0, n, 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)}
%o (PARI) /* (1) G.f. e_q(x,q) / E_q(x,q) at q=-x: */
%o {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)}
%o (PARI) /* (1) G.f. e_q(x,q) / E_q(x,q) at q=-x: */
%o {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)}
%o (PARI) /* (2) G.f. exp( Sum_{n>=1} (1+x)^(2*n)/(1-x^(2*n)) * x^(2*n)/n): */
%o {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)}
%o (PARI) /* (3) G.f. Product_{n>=1} 1/(1 - x^(2*n)*(1+x)^2): */
%o {a(n)=polcoeff(1/prod(k=1, n, 1-(1+x)^2*x^(2*k)+x*O(x^n)), n)}
%Y Cf. A198199 (dual), A152398 (e_q), A198197 (E_q), A198201 (q-cosh), A198202 (q-sinh), A198242 (q-Cosh), A198243 (q-Sinh).
%K nonn
%O 0,4
%A _Paul D. Hanna_, Oct 22 2011
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