A204272 a(n) = sigma_2(n)*Pell(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n.

1, 10, 50, 252, 754, 3500, 8450, 34680, 89635, 309140, 700402, 2910600, 5688370, 20195500, 50706500, 160553712, 329639810, 1248615550, 2398289458, 8732957688, 19306982500, 56865638380, 119281100930, 461838762000, 853941516771

Offset

1,2

Comments

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^2*x^n/(1-x^n) = Sum_{n>=1} sigma_2(n)*x^n.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{n>=1} n^2*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_2(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

Example

G.f.: A(x) = x + 10*x^2 + 50*x^3 + 252*x^4 + 754*x^5 + 3500*x^6 +...
where A(x) = x/(1-2*x-x^2) + 2^2*2*x^2/(1-6*x^2+x^4) + 3^2*5*x^3/(1-14*x^3-x^6) + 4^2*12*x^4/(1-34*x^4+x^8) + 5^2*29*x^5/(1-82*x^5-x^10) + 6^2*70*x^6/(1-198*x^6+x^12) +...+ n^2*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Mathematica

With[{nn=30}, Times@@@Thread[{Rest[LinearRecurrence[{2, 1}, {0, 1}, nn+1]], DivisorSigma[ 2, Range[nn]]}]] (* Harvey P. Dale, Oct 21 2015 *)

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

Crossrefs

Cf. A203849, A204270, A204271, A204273, A204274, A204275, A001157 (sigma_2), A002203, A000045.

Sequence in context: A201210 A003207 A095687 * A220149 A154410 A060156

Adjacent sequences: A204269 A204270 A204271 * A204273 A204274 A204275

Keyword

nonn

Author

Paul D. Hanna, Jan 14 2012

Status

approved