cp's OEIS Frontend

A205973 a(n) = Fibonacci(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.

%I A205973 #7 Mar 30 2012 18:37:34
%S A205973 1,-9,27,-18,-351,1080,216,-5850,9639,-306,-35640,96120,-16848,
%T A205973 -356490,508950,131760,-1821015,4139424,69768,-13621698,18996120,
%U A205973 -4925700,-57383640,136178064,21282912,-405810225,557193870,-1767762,-1859194350,3887571240,-539161920
%N A205973 a(n) = Fibonacci(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.
%C A205973 Compare the g.f. to the Lambert series of A109041:
%C A205973 1 - 9*Sum_{n>=1} Kronecker(n,3)*n^2*x^n/(1-x^n).
%F A205973 G.f.: 1 - 9*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*n^2*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
%e A205973 G.f.: A(x) = 1 - 9*x + 27*x^2 - 18*x^3 - 351*x^4 + 1080*x^5 + 216*x^6 +...
%e A205973 where A(x) = 1 - 1*9*x + 1*27*x^2 - 2*9*x^3 - 3*117*x^4 + 5*216*x^5 + 8*27*x^6 - 13*450*x^7 + 21*459*x^8 +...+ Fibonacci(n)*A109041(n)*^n +...
%e A205973 The g.f. is also given by the identity:
%e A205973 A(x) = 1 - 9*( 1*1*x/(1-x-x^2) - 1*4*x^2/(1-3*x^2+x^4) + 3*16*x^4/(1-7*x^4+x^8) - 5*25*x^5/(1-11*x^5-x^10) + 13*49*x^7/(1-29*x^7-x^14) - 21*64*x^8/(1-47*x^8+x^16) +...).
%e A205973 The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
%o A205973 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205973 {a(n)=polcoeff(1 - 9*sum(m=1,n,fibonacci(m)*kronecker(m,3)*m^2*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o A205973 for(n=0,40,print1(a(n),", "))
%Y A205973 Cf. A109041, A205972, A205974, A203847, A000204 (Lucas).
%Y A205973 Cf. A209453 (Pell variant).
%K A205973 sign
%O A205973 0,2
%A A205973 _Paul D. Hanna_, Feb 04 2012