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

Author

Paul D. Hanna, Jan 14 2012

Keywords

• nonn

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.

Examples

			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)) +...
		

Links

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

Crossrefs

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

Programs

Mathematica

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

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)}

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.