login
A203804
G.f.: exp( Sum_{n>=1} A000204(n)^4 * x^n/n ) where A000204 is the Lucas numbers.
11
1, 1, 41, 126, 1526, 7854, 63629, 400789, 2870629, 19254504, 133376760, 909578760, 6249172910, 42785312510, 293403088510, 2010553849020, 13781960765020, 94458627485820, 647442212896270, 4437595353800270, 30415849505902910, 208472981440853160, 1428896115173689560
OFFSET
0,3
COMMENTS
More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).
LINKS
FORMULA
G.f.: 1/( (1-x)^6 * (1+3*x+x^2)^4 * (1-7*x+x^2) ).
G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203854(n) where A203854(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^3.
EXAMPLE
G.f.: A(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 7854*x^5 + 63629*x^6 +...
where
log(A(x)) = x + 3^4*x^2/2 + 4^4*x^3/3 + 7^4*x^4/4 + 11^4*x^5/5 + 18^4*x^6/6 + 29^4*x^7/7 + 47^4*x^8/8 +...+ Lucas(n)^4*x^n/n +...
MATHEMATICA
CoefficientList[Series[1/((1 - x)^6*(1 + 3*x + x^2)^4*(1 - 7*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 24 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)^4*x^k/k)+x*O(x^n)), n)}
(PARI) {a(n, m=2)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m, m) * prod(k=1, m, 1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m, m-k)), n)}
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 06 2012
STATUS
approved