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A204273 a(n) = sigma_3(n)*Pell(n), where sigma_3(n) = A001158(n), the sum of cubes of divisors of n.

%I A204273 #11 Jan 06 2018 22:05:28
%S A204273 1,18,140,876,3654,17640,58136,238680,745645,2696652,7647012,28329840,
%T A204273 73547278,250101072,688048200,2203964592,5585689746,18696302730,
%U A204273 45448247740,147116748744,371929710880,1117549627704,2738514030408,8899904613600
%N A204273 a(n) = sigma_3(n)*Pell(n), where sigma_3(n) = A001158(n), the sum of cubes of divisors of n.
%C A204273 Compare g.f. to the Lambert series identity: Sum_{n>=1} n^3*x^n/(1-x^n) = Sum_{n>=1} sigma_3(n)*x^n.
%H A204273 G. C. Greubel, <a href="/A204273/b204273.txt">Table of n, a(n) for n = 1..1000</a>
%F A204273 G.f.: Sum_{n>=1} n^3*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_3(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.
%e A204273 G.f.: A(x) = x + 18*x^2 + 140*x^3 + 876*x^4 + 3654*x^5 + 17640*x^6 + ...
%e A204273 where A(x) = x/(1-2*x-x^2) + 2^3*2*x^2/(1-6*x^2+x^4) + 3^3*5*x^3/(1-14*x^3-x^6) + 4^3*12*x^4/(1-34*x^4+x^8) + 5^3*29*x^5/(1-82*x^5-x^10) + 6^3*70*x^6/(1-198*x^6+x^12) + ... + n^3*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) + ...
%t A204273 Table[DivisorSigma[3, n] Fibonacci[n, 2], {n, 1, 50}] (* _G. C. Greubel_, Jan 05 2018 *)
%o A204273 (PARI) /* Subroutines used in PARI programs below: */
%o A204273 {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
%o A204273 {A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}
%o A204273 (PARI) {a(n)=sigma(n,3)*Pell(n)}
%o A204273 (PARI) {a(n)=polcoeff(sum(m=1,n,m^3*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
%Y A204273 Cf. A203838, A204270, A204271, A204272, A204274, A204275, A001158 (sigma_3), A002203, A000045.
%K A204273 nonn
%O A204273 1,2
%A A204273 _Paul D. Hanna_, Jan 14 2012