A225525 a(n) = (sigma(2*n) - sigma(n))*Lucas(n) where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.

2, 12, 32, 56, 132, 288, 464, 752, 1976, 2952, 4776, 10304, 14588, 26976, 65472, 70624, 128556, 300456, 373960, 726096, 1566464, 1900944, 3075792, 6635648, 10401182, 15200808, 35136320, 45481408, 68991060, 178607808, 192662336, 311734208, 756594816, 918147096, 1980790944, 3472069328

Offset: 1

Author

Paul D. Hanna, May 09 2013

Keywords

• nonn

Comments

Compare l.g.f. to log(theta_4(x)) = Sum_{n>=1} (sigma(2*n)-sigma(n))*x^n/n, where Jacobi theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n*x^(n^2).

Examples

			L.g.f.: L(x) = 2*x + 4*3*x^2/2 + 8*4*x^3/3 + 8*7*x^4/4 + 12*11*x^5/5 + 16*18*x^6/6 +...
where
exp(-L(x)) = 1 - 2*x - 4*x^2 + 14*x^4 + 16*x^5 + 4*x^8 - 152*x^9 - 188*x^10 +...+ A203850(n)*x^n +...
Also,
exp(L(x)) = 1 + 2*x + 8*x^2 + 24*x^3 + 66*x^4 + 184*x^5 + 488*x^6 + 1248*x^7 +...+ A225524(n)*x^n +...
Exponentiation yields the product:
exp(L(x)) = (1+x-x^2)/(1-x-x^2) * (1+3*x^2+x^4)/(1-3*x^2+x^4) * (1+4*x^3-x^6)/(1-4*x^3-x^6) * (1+7*x^4+x^8)/(1-7*x^4+x^8) * (1+11*x^5-x^10)/(1-11*x^5-x^10) *...* (1 + Lucas(n)*x^n + (-x^2)^n)/(1 - Lucas(n)*x^n + (-x^2)^n) *...
		

Crossrefs

Cf. A203850, A225524.

Programs

Mathematica

Table[(DivisorSigma[1,2n]-DivisorSigma[1,n])LucasL[n],{n,40}] (* Harvey P. Dale, Sep 10 2016 *)

PARI

{a(n)=(sigma(2*n) - sigma(n))*(fibonacci(n-1)+fibonacci(n+1))}
for(n=1,40,print1(a(n),", "))

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(-log(prod(m=1, n\2+1, (1 - Lucas(2*m-1)*x^(2*m-1) - x^(4*m-2))^2*(1 - Lucas(2*m)*x^(2*m) + x^(4*m) +x*O(x^n)))), n)}
for(n=1,40,print1(a(n),", "))

Formula

Logarithmic derivative of A225524 and A203850 (up to sign).

L.g.f.: Sum_{n>=1} log( (1 + Lucas(n)*x^n + (-x^2)^n) / (1 - Lucas(n)*x^n + (-x^2)^n) ) = Sum_{n>=1} a(n)*x^n/n.

a(n) == 0 (mod 2); a(n) == 2 (mod 4) iff n is congruent to 1 or 5 mod 6 (A007310).