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%I #10 Jan 07 2018 04:14:00
%S 1,4,10,36,58,280,338,1632,2955,9512,11482,83160,66922,323128,780100,
%T 2354160,2273378,16465260,13250218,95966568,154455860,372889432,
%U 450117362,4346717760,3935214363,12667263848,30581480180,110745336312,89120964298
%N a(n) = tau(n)*Pell(n), where tau(n) = A000005(n), the number of divisors of n.
%C Compare g.f. to the Lambert series identity: Sum_{n>=1} x^n/(1-x^n) = Sum_{n>=1} tau(n)*x^n.
%C Related identities:
%C (1) Sum_{n>=1} n^k*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_{k}(n)*Pell(n)*x^n for k>=0.
%C (2) Sum_{n>=1} phi(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Pell(n)*x^n.
%C (3) Sum_{n>=1} moebius(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = x.
%C (4) Sum_{n>=1} lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Pell(n^2)*x^(n^2).
%H G. C. Greubel, <a href="/A204270/b204270.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: Sum_{n>=1} Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} tau(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.
%e G.f.: A(x) = 1 + 4*x + 10*x^2 + 36*x^3 + 58*x^4 + 280*x^5 + 338*x^6 +...
%e where A(x) = 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) +...+ Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...
%t Table[DivisorSigma[0, n] Fibonacci[n, 2], {n, 1, 50}] (* _G. C. Greubel_, Jan 05 2018 *)
%o (PARI) /* Subroutines used in PARI programs below: */
%o {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
%o {A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}
%o (PARI) {a(n)=sigma(n,0)*Pell(n)}
%o (PARI) {a(n)=polcoeff(sum(m=1,n,Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
%Y Cf. A203847, A204271, A204272, A204273, A204274, A204275, A000005 (tau), A002203, A000045.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 14 2012
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