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 A209454 a(n) = Pell(n)*A033719(n) for n>=1, with a(0)=1, where A033719 lists the coefficients in theta_3(q)*theta_3(q^7). 4

%I

%S 1,2,0,0,24,0,0,338,1632,1970,0,22964,0,0,0,0,2824992,0,0,0,0,0,0,

%T 900234724,0,2623476242,0,0,36915112104,178241928596,0,0,

%U 5016108528384,0,0,0,42600007379160,205691031143924,0,0,0,0,0,40725785296405556,98320743200877072,0,0,0,0

%N a(n) = Pell(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 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-(-x)^n) (the Lambert series of A033719).

%H G. C. Greubel, <a href="/A209454/b209454.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1 + 2*Sum_{n>=1} Pell(n)*Kronecker(n,7)*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 + 2*x + 24*x^4 + 338*x^7 + 1632*x^8 + 1970*x^9 + 22964*x^11 +...

%e where A(x) = 1 + 1*2*x + 12*2*x^4 + 169*2*x^7 + 408*4*x^8 + 985*2*x^9 + 5741*4*x^11 + 470832*6*x^16 + 225058681*4*x^23 +...+ Pell(n)*A033719(n)*x^n +...

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

%e A(x) = 1 + 2*( 1*x/(1+2*x-x^2) + 2*x^2/(1-6*x^2+x^4) - 5*x^3/(1+14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) - 70*x^6/(1-198*x^6+x^12) + 0*169*13*x^7/(1+478*x^7-x^14) +...).

%e The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].

%t A033719[n_]:= SeriesCoefficient[EllipticTheta[3, 0, x] EllipticTheta[3, 0, x^7], {x, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A033719[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 + 2*sum(m=1,n,Pell(m)*kronecker(m,7)*x^m/(1-A002203(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}

%o for(n=0,50,print1(a(n),", "))

%Y Cf. A033719, A205974, A209453, A209455, A204270, A000129 (Pell), A002203.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 10 2012

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Last modified July 25 13:05 EDT 2021. Contains 346290 sequences. (Running on oeis4.)