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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.
5
1, 18, 140, 876, 3654, 17640, 58136, 238680, 745645, 2696652, 7647012, 28329840, 73547278, 250101072, 688048200, 2203964592, 5585689746, 18696302730, 45448247740, 147116748744, 371929710880, 1117549627704, 2738514030408, 8899904613600
OFFSET
1,2
COMMENTS
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.
LINKS
FORMULA
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.
EXAMPLE
G.f.: A(x) = x + 18*x^2 + 140*x^3 + 876*x^4 + 3654*x^5 + 17640*x^6 + ...
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)) + ...
MATHEMATICA
Table[DivisorSigma[3, n] Fibonacci[n, 2], {n, 1, 50}] (* G. C. Greubel, Jan 05 2018 *)
PROG
(PARI) /* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}
(PARI) {a(n)=sigma(n, 3)*Pell(n)}
(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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 14 2012
STATUS
approved