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%I #13 Jan 04 2018 01:35:21
%S 1,-6,24,-30,-72,0,840,-2028,4896,-5910,0,0,-83160,-401532,1938768,0,
%T -2824992,0,32930520,-79501308,0,-463367580,0,0,6520076640,
%U -7870428726,76003583088,-45872220270,-221490672624,0,0,-3116610274188,7524162792576,0,0,0,-127800022137480
%N a(n) = Pell(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.
%C Compare the g.f. to the Lambert series of A122859: 1 - 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1+x^n).
%H G. C. Greubel, <a href="/A209452/b209452.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1 - 6*Sum_{n>=1} Pell(n)*Kronecker(n,3)*x^n/(1 + A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
%e G.f.: A(x) = 1 - 6*x + 24*x^2 - 30*x^3 - 72*x^4 + 840*x^6 - 2028*x^7 + ...
%e where A(x) = 1 - 1*6*x + 2*12*x^2 - 5*6*x^3 - 12*6*x^4 + 70*12*x^6 - 169*12*x^7 + 408*12*x^8 - 985*6*x^9 + ... + Pell(n)*A122859(n)*x^n + ...
%e The g.f. is also given by the identity:
%e A(x) = 1 - 6*( 1*x/(1+2*x-x^2) - 2*x^2/(1+6*x^2+x^4) + 12*x^4/(1+34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 408*x^8/(1+1154*x^8+x^16) + ...).
%e The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
%t A122859[n_]:= SeriesCoefficient[EllipticTheta[4, 0, q]^3/EllipticTheta[4, 0, q^3], {q, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A122859[n], {n, 1, 50}]] (* _G. C. Greubel_, Jan 02 2017 *)
%o (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
%o {A002203(n)=Pell(n-1)+Pell(n+1)}
%o {a(n)=polcoeff(1 - 6*sum(m=1,n,Pell(m)*kronecker(m,3)*x^m/(1+A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A122859, A205972, A209451, A209453, A209446, A209449, A204270, A000129 (Pell), A002203.
%K sign
%O 0,2
%A _Paul D. Hanna_, Mar 10 2012
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