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%I #8 Jan 03 2018 17:55:11
%S 1,4,24,80,336,696,3360,676,24480,51220,171216,275568,1552320,1873816,
%T 969384,18722400,58383168,81841608,428096760,530008720,2687063904,
%U 617823440,13424019552,21605633376,130401532800,162655527004,532025081616,1223259207200
%N a(n) = Pell(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.
%H G. C. Greubel, <a href="/A209456/b209456.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1 + 4*Sum_{n>=1} Pell(n)*Chi(n,7)*n*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 + 4*x + 24*x^2 + 80*x^3 + 336*x^4 + 696*x^5 + 3360*x^6 +...
%e where A(x) = 1 + 1*4*x + 2*12*x^2 + 5*16*x^3 + 12*28*x^4 + 29*24*x^5 + 70*48*x^6 + 169*4*x^7 + 408*60*x^8 + 985*52*x^9 +...+ Pell(n)*A028594(n)*x^n +...
%e The g.f. is also given by the identity:
%e A(x) = 1 + 4*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 5*3*x^3/(1-14*x^3-x^6) + 12*4*x^4/(1-34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 70*6*x^6/(1-198*x^6+x^12) + 0*169*7*x^7/(1+478*x^7-x^14) +...).
%e The values of the Dirichlet character Chi(n,7) repeat [1,1,1,1,1,1,0, ...].
%t A028594[n_]:= If[n < 1, Boole[n == 0], 4*Sum[If[Mod[d, 7] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A028594[n], {n,1,50}]] (* _G. C. Greubel_, Jan 03 2018 *)
%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 + 4*sum(m=1,n,Pell(m)*kronecker(m,7)^2*m*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o for(n=0,35,print1(a(n),", "))
%Y Cf. A028594, A205976, A209455, A204270, A000129 (Pell), A002203.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 10 2012
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