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%I #7 Sep 10 2016 13:43:36
%S 2,12,32,56,132,288,464,752,1976,2952,4776,10304,14588,26976,65472,
%T 70624,128556,300456,373960,726096,1566464,1900944,3075792,6635648,
%U 10401182,15200808,35136320,45481408,68991060,178607808,192662336,311734208,756594816,918147096,1980790944,3472069328
%N 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.
%C 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).
%F Logarithmic derivative of A225524 and A203850 (up to sign).
%F 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.
%F a(n) == 0 (mod 2); a(n) == 2 (mod 4) iff n is congruent to 1 or 5 mod 6 (A007310).
%e 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 +...
%e where
%e 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 +...
%e Also,
%e 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 +...
%e Exponentiation yields the product:
%e 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) *...
%t Table[(DivisorSigma[1,2n]-DivisorSigma[1,n])LucasL[n],{n,40}] (* _Harvey P. Dale_, Sep 10 2016 *)
%o (PARI) {a(n)=(sigma(2*n) - sigma(n))*(fibonacci(n-1)+fibonacci(n+1))}
%o for(n=1,40,print1(a(n),", "))
%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o {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)}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A203850, A225524.
%K nonn
%O 1,1
%A _Paul D. Hanna_, May 09 2013
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