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A209453 a(n) = Pell(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.

%I A209453 #8 Jan 03 2018 17:55:04
%S A209453 1,-9,54,-45,-1404,6264,1890,-76050,187272,-8865,-1540944,6200280,
%T A209453 -1621620,-51195330,109055700,42125400,-868685040,2946297888,74093670,
%U A209453 -21584605122,44912353824,-17376284250,-302040439920,1069478852112,249392931480,-7095191496489
%N A209453 a(n) = Pell(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.
%C A209453 Compare the g.f. to the Lambert series of A109041:
%C A209453 1 - 9*Sum_{n>=1} Kronecker(n,3)*n^2*x^n/(1-x^n).
%H A209453 G. C. Greubel, <a href="/A209453/b209453.txt">Table of n, a(n) for n = 0..1000</a>
%F A209453 G.f.: 1 - 9*Sum_{n>=1} Pell(n)*Kronecker(n,3)*n^2*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
%e A209453 G.f.: A(x) = 1 - 9*x + 54*x^2 - 45*x^3 - 1404*x^4 + 6264*x^5 + 1890*x^6 +...
%e A209453 where A(x) = 1 - 1*9*x + 2*27*x^2 - 5*9*x^3 - 12*117*x^4 + 29*216*x^5 + 70*27*x^6 - 169*450*x^7 + 408*459*x^8 +...+ Pell(n)*A109041(n)*^n +...
%e A209453 The g.f. is also given by the identity:
%e A209453 A(x) = 1 - 9*( 1*1*x/(1-2*x-x^2) - 2*4*x^2/(1-6*x^2+x^4) + 12*16*x^4/(1-34*x^4+x^8) - 29*25*x^5/(1-82*x^5-x^10) + 169*49*x^7/(1-478*x^7-x^14) - 408*64*x^8/(1-1154*x^8+x^16) +...).
%e A209453 The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
%t A209453 A109041[n_]:= If[n < 1, Boole[n == 0], -9 DivisorSum[n, #^2 KroneckerSymbol[-3, #] &]]; Join[{1}, Table[Fibonacci[n, 2]*A109041[n], {n, 1, 50}]] (* _G. C. Greubel_, Jan 02 2018 *)
%o A209453 (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
%o A209453 {A002203(n)=Pell(n-1)+Pell(n+1)}
%o A209453 {a(n)=polcoeff(1 - 9*sum(m=1,n,Pell(m)*kronecker(m,3)*m^2*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o A209453 for(n=0,40,print1(a(n),", "))
%Y A209453 Cf. A109041, A205973, A209452, A209454, A204270, A000129 (Pell), A002203.
%K A209453 sign
%O A209453 0,2
%A A209453 _Paul D. Hanna_, Mar 10 2012