cp's OEIS Frontend

A198199 G.f.: q-Cosh(x,q)^2 - q-Sinh(x,q)^2 at q=-x.

%I A198199 #19 Feb 16 2025 08:33:15
%S A198199 1,0,-1,-2,-2,-2,-1,2,5,6,7,10,12,8,-1,-8,-11,-16,-26,-38,-50,-58,-52,
%T A198199 -30,-8,0,4,24,63,112,170,232,268,252,208,186,193,190,154,92,-5,-164,
%U A198199 -383,-630,-873,-1062,-1128,-1080,-1055,-1172,-1374,-1508,-1508,-1392,-1139,-700,-85,684,1627,2652,3455
%N A198199 G.f.: q-Cosh(x,q)^2 - q-Sinh(x,q)^2 at q=-x.
%C A198199 This sequence illustrates in part the identities:
%C A198199 * q-Cosh(x,q)^2 - q-Sinh(x,q)^2 = E_q(x,q) / e_q(x,q),
%C A198199 * q-cosh(x,q)^2 - q-sinh(x,q)^2 = e_q(x,q) / E_q(x,q).
%C A198199 Here the following q-analogs are employed (see MathWorld links):
%C A198199 q-cosh(x,q) = Sum_{n>=0} x^(2*n)/faq(2*n,q),
%C A198199 q-sinh(x,q) = Sum_{n>=0} x^(2*n+1)/faq(2*n+1,q),
%C A198199 and the dual expressions:
%C A198199 q-Cosh(x,q) = Sum_{n>=0} q^(n*(2*n-1))*x^(2*n)/faq(2*n,q),
%C A198199 q-Sinh(x,q) = Sum_{n>=0} q^(n*(2*n+1))*x^(2*n+1)/faq(2*n+1,q),
%C A198199 along with the dual q-exponential functions of x:
%C A198199 e_q(x,q) = Sum_{n>=0} x^n/faq(n,q),
%C A198199 E_q(x,q) = Sum_{n>=0} q^(n*(n-1)/2) * x^n/faq(n,q),
%C A198199 where
%C A198199 faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.
%H A198199 Paul D. Hanna, <a href="/A198199/b198199.txt">Table of n, a(n) for n = 0..1000</a>
%H A198199 Eric Weisstein, <a href="https://mathworld.wolfram.com/q-ExponentialFunction.html">q-Exponential Function</a> from MathWorld.
%H A198199 Eric Weisstein, <a href="https://mathworld.wolfram.com/q-Cosine.html">q-Cosine Function</a> from MathWorld.
%H A198199 Eric Weisstein, <a href="https://mathworld.wolfram.com/q-Sine.html">q-Sine Function</a> from MathWorld.
%H A198199 Eric Weisstein, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a> from MathWorld.
%F A198199 (1) G.f.: E_q(x,q) / e_q(x,q) at q=-x, where
%F A198199 e_q(x,-x) = Sum_{n>=0} x^n/Product(k=1, n, (1 - (-x)^k)/(1+x)),
%F A198199 E_q(x,-x) = Sum_{n>=0} (-x)^(n*(n-1)/2) * x^n/Product(k=1, n, (1 - (-x)^k)/(1+x)).
%F A198199 (2) G.f.: exp( -Sum_{n>=1} (1+x)^(2*n)/(1-x^(2*n)) * x^(2*n)/n ).
%F A198199 (3) G.f.: Product_{n>=1} (1 - x^(2*n)*(1+x)^2).
%F A198199 (4) Given g.f. A(x), A( (sqrt(5)-1)/2 ) = 0.
%e A198199 G.f.: A(x) = 1 - x^2 - 2*x^3 - 2*x^4 - 2*x^5 - x^6 + 2*x^7 + 5*x^8 +...
%e A198199 The g.f. may be expressed by:
%e A198199 (0) A(x) = q-Cosh(x,q)^2 - q-Sinh(x,q)^2 at q=-x, where
%e A198199 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 +...
%e A198199 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 +...
%e A198199 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 +...
%e A198199 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 +...
%e A198199 (1) A(x) = E_q(x,q) / e_q(x,q) at q=-x, where
%e A198199 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 A198199 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 A198199 (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 A198199 (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) *...
%o A198199 (PARI) /* (0) G.f. q-Cosh(x,q)^2 - q-Sinh(x,q)^2 at q=-x: */
%o A198199 {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)}
%o A198199 (PARI) /* (1) G.f. E_q(x,q) / e_q(x,q) at q=-x: */
%o A198199 {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 A198199 (PARI) /* (1) G.f. E_q(x,q) / e_q(x,q) at q=-x: */
%o A198199 {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 A198199 (PARI) /* (2) G.f. exp( -Sum_{n>=1} (1+x)^(2*n)/(1-x^(2*n)) * x^(2*n)/n): */
%o A198199 {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 A198199 (PARI) /* (3) G.f. Product_{n>=1} (1 - x^(2*n)*(1+x)^2): */
%o A198199 {a(n)=polcoeff(prod(k=1, n, 1-(1+x)^2*x^(2*k)+x*O(x^n)), n)}
%Y A198199 Cf. A198200 (dual), A152398 (e_q), A198197 (E_q), A198242 (q-Cosh), A198243 (q-Sinh), A198201 (q-cosh), A198202 (q-sinh).
%K A198199 sign
%O A198199 0,4
%A A198199 _Paul D. Hanna_, Oct 22 2011