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%I #7 Mar 30 2012 18:37:34
%S 1,2,0,0,6,0,0,26,84,68,0,356,0,0,0,0,5922,0,0,0,0,0,0,114628,0,
%T 150050,0,0,635622,2056916,0,0,17426472,0,0,0,29860704,96631268,0,0,0,
%U 0,0,1733977748,2805634932,0,0,0,0,15557484098,0,0,0,213265164692,0,0
%N a(n) = Fibonacci(n)*A033719(n) for n>=1, with a(0)=1, where A033719 lists the coefficients in theta_3(q)*theta_3(q^7).
%C Compare g.f. to the Lambert series of A033719:
%C 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-(-x)^n).
%F G.f.: 1 + 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,7)*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).
%e G.f.: A(x) = 1 + 2*x + 6*x^4 + 26*x^7 + 84*x^8 + 68*x^9 + 356*x^11 +...
%e where A(x) = 1 + 1*2*x + 3*2*x^4 + 13*2*x^7 + 21*4*x^8 + 34*2*x^9 + 89*4*x^11 + 987*6*x^16 + 28657*4*x^23 +...+ Fibonacci(n)*A033719(n)*x^n +...
%e The g.f. is also given by the identity:
%e A(x) = 1 + 2*( 1*x/(1+x-x^2) + 1*x^2/(1-3*x^2+x^4) - 2*x^3/(1+4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) - 5*x^5/(1+11*x^5-x^10) - 8*x^6/(1-18*x^6+x^12) + 0*13*x^7/(1+29*x^7-x^14) +...).
%e The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].
%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o {a(n)=polcoeff(1 + 2*sum(m=1,n,fibonacci(m)*kronecker(m,7)*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A033719, A205973, A205975, A203847, A000204 (Lucas).
%Y Cf. A209454 (Pell variant).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 04 2012
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