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A204271 a(n) = sigma(n)*Pell(n), where sigma(n) = A000203(n), the sum of divisors of n. 5
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 (list; graph; refs; listen; history; text; internal format)
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: A058494 A147979 A118265 * A255469 A226638 A274071

Adjacent sequences:  A204268 A204269 A204270 * A204272 A204273 A204274

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 14 2012

STATUS

approved

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Last modified April 26 08:05 EDT 2019. Contains 322472 sequences. (Running on oeis4.)