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A204270
a(n) = tau(n)*Pell(n), where tau(n) = A000005(n), the number of divisors of n.
23
1, 4, 10, 36, 58, 280, 338, 1632, 2955, 9512, 11482, 83160, 66922, 323128, 780100, 2354160, 2273378, 16465260, 13250218, 95966568, 154455860, 372889432, 450117362, 4346717760, 3935214363, 12667263848, 30581480180, 110745336312, 89120964298
OFFSET
1,2
COMMENTS
Compare g.f. to the Lambert series identity: Sum_{n>=1} x^n/(1-x^n) = Sum_{n>=1} tau(n)*x^n.
Related identities:
(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.
(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.
(3) Sum_{n>=1} moebius(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = x.
(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).
LINKS
FORMULA
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.
EXAMPLE
G.f.: A(x) = 1 + 4*x + 10*x^2 + 36*x^3 + 58*x^4 + 280*x^5 + 338*x^6 +...
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)) +...
MATHEMATICA
Table[DivisorSigma[0, 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, 0)*Pell(n)}
(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)}
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 14 2012
STATUS
approved