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a(n) = sigma(n)*Lucas(n) where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.
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%I #9 May 09 2013 23:37:08

%S 1,9,16,49,66,216,232,705,988,2214,2388,9016,7294,20232,32736,68417,

%T 64278,225342,186980,635334,783232,1425708,1537896,6220920,5200591,

%U 11400606,17568160,39796232,34495530,133955856,96331168,306863361,378297408,688610322,990395472,3038060662

%N a(n) = sigma(n)*Lucas(n) where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.

%H Paul D. Hanna, <a href="/A225528/b225528.txt">Table of n, a(n) for n = 1..1000</a>

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

%F Logarithmic derivative of A156234.

%e L.g.f.: L(x) = x + 9*x^2/2 + 16*x^3/3 + 49*x^4/4 + 66*x^5/5 + 216*x^6/6 +...

%e which is equivalent to:

%e L(x) = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 + 8*29*x^7/7 + 15*47*x^8/8 +...+ sigma(n)*Lucas(n)*x^n/n +...

%e where exponentiation yields the g.f. of A156234:

%e exp(L(x)) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 63*x^5 + 170*x^6 + 355*x^7 +...+ A156234(n)*x^n +...

%e and equals the product:

%e exp(L(x)) = 1/((1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6) * (1-7*x^4+x^8) * (1-11*x^5-x^10) * (1-18*x^6+x^12) *...* (1 - Lucas(n)*x^n + (-x^2)^n) *...).

%o (PARI) {a(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(sum(m=1, n, -log(1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))), n)}

%o for(n=1,40,print1(a(n),", "))

%Y Cf. A156234, A225525.

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 09 2013