A204271 a(n) = sigma(n)*Pell(n), where sigma(n) = A000203(n), the sum of divisors of n.

1, 6, 20, 84, 174, 840, 1352, 6120, 12805, 42804, 68892, 388080, 468454, 1938768, 4680600, 14595792, 20460402, 107024190, 132502180, 671765976, 1235646880, 3356004888, 5401408344, 32600383200, 40663881751, 133006270404, 305814801800

Offset

1,2

Comments

Compare g.f. to the Lambert series identity: Sum_{n>=1} n*x^n/(1-x^n) = Sum_{n>=1} sigma(n)*x^n.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{n>=1} n*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

Example

G.f.: A(x) = x + 6*x^2 + 20*x^3 + 84*x^4 + 174*x^5 + 840*x^6 + 1352*x^7 +...
where A(x) = x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 3*5*x^3/(1-14*x^3-x^6) + 4*12*x^4/(1-34*x^4+x^8) + 5*29*x^5/(1-82*x^5-x^10) + 6*70*x^6/(1-198*x^6+x^12) +...+ n*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Mathematica

Table[DivisorSigma[1, 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)*Pell(n)}
(PARI) {a(n)=polcoeff(sum(m=1, n, m*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}

Crossrefs

Cf. A000203 (sigma), A000129, A203848, A204270, A204272, A204273, A204274, A204275, A002203, A000045.

Sequence in context: A147979 A330245 A118265 * A255469 A226638 A274071

Adjacent sequences: A204268 A204269 A204270 * A204272 A204273 A204274

Keyword

nonn

Author

Paul D. Hanna, Jan 14 2012

Status

approved