OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1000
FORMULA
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.
Logarithmic derivative of A156234.
EXAMPLE
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 +...
which is equivalent to:
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 +...
where exponentiation yields the g.f. of A156234:
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 +...
and equals the product:
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) *...).
PROG
(PARI) {a(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(sum(m=1, n, -log(1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 09 2013
STATUS
approved