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A203805
G.f.: exp( Sum_{n>=1} A000204(n)^5 * x^n/n ) where A000204 is the Lucas numbers.
9
1, 1, 122, 463, 11985, 85456, 1262166, 12018742, 145326748, 1540766090, 17495016342, 191731126832, 2138972609189, 23652975370501, 262682339212290, 2911255335387883, 32296421465575573, 358120616523262016, 3971885483375619384, 44047530724737577400
OFFSET
0,3
COMMENTS
More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).
LINKS
FORMULA
G.f.: 1/( (1-x-x^2)^10 * (1+4*x-x^2)^5 * (1-11*x-x^2) ).
G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203855(n) where A203855(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^4.
EXAMPLE
G.f.: A(x) = 1 + x + 122*x^2 + 463*x^3 + 11985*x^4 + 85456*x^5 + ...
where
log(A(x)) = x + 3^5*x^2/2 + 4^5*x^3/3 + 7^5*x^4/4 + 11^5*x^5/5 + 18^5*x^6/6 + 29^5*x^7/7 + 47^5*x^8/8 + ... + Lucas(n)^5*x^n/n + ...
MATHEMATICA
CoefficientList[Series[1/((1 - x - x^2)^10*(1 + 4*x - x^2)^5*(1 - 11*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)
PROG
(PARI) /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
(PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^5*x^k/k)+x*O(x^n)), n)}
(PARI) {a(n, m=2)=polcoeff(prod(k=0, m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1, m-k)), n)}
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 06 2012
STATUS
approved