A205976 - OEIS

%I #7 Mar 30 2012 18:37:34

%S 1,4,12,32,84,120,384,52,1260,1768,3960,4272,16128,13048,4524,58560,

%T 122388,114984,403104,334480,1136520,175136,2550384,2751072,11128320,

%U 9303100,20394024,31426880,8898708,61707480,239627520,172322432,548933868,676718976,1231823592

%N a(n) = Fibonacci(n)*A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in  (theta_3(x)*theta_3(7*x)+theta_2(x)*theta_2(7*x))^2.

%C Compare g.f. to the Lambert series of A028594:

%C 1 + 4*Sum_{n>=1} Chi(n,7)*n*x^n/(1-x^n).

%C Here Chi(n,7) = principal Dirichlet character of n modulo 7.

%F G.f.: 1 + 4*Sum_{n>=1} Fibonacci(n)*Chi(n,7)*n*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

%e G.f.: A(x) = 1 + 4*x + 12*x^2 + 32*x^3 + 84*x^4 + 120*x^5 + 384*x^6 + 52*x^7 +...

%e where A(x) = 1 + 1*4*x + 1*12*x^2 + 2*16*x^3 + 3*28*x^4 + 5*24*x^5 + 8*48*x^6 + 13*4*x^7 + 21*60*x^8 + 34*52*x^9 +...+ Fibonacci(n)*A028594(n)*x^n +...

%e The g.f. is also given by the identity:

%e A(x) = 1 + 4*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1-3*x^2+x^4) + 2*3*x^3/(1-4*x^3-x^6) + 3*4*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 8*6*x^6/(1-18*x^6+x^12) + 0*13*7*x^7/(1+29*x^7-x^14) +...).

%e The values of the Dirichlet character Chi(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 + 4*sum(m=1,n,fibonacci(m)*kronecker(m,7)^2*m*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}

%o for(n=0,60,print1(a(n),", "))

%Y Cf. A028594, A205975, A203847, A000204 (Lucas).

%Y Cf. A209456 (Pell variant).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 04 2012