A203805 - OEIS

%I #13 Dec 26 2017 03:15:06

%S 1,1,122,463,11985,85456,1262166,12018742,145326748,1540766090,

%T 17495016342,191731126832,2138972609189,23652975370501,

%U 262682339212290,2911255335387883,32296421465575573,358120616523262016,3971885483375619384,44047530724737577400

%N G.f.: exp( Sum_{n>=1} A000204(n)^5 * x^n/n ) where A000204 is the Lucas numbers.

%C 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).

%H G. C. Greubel, <a href="/A203805/b203805.txt">Table of n, a(n) for n = 0..950</a>

%F G.f.: 1/( (1-x-x^2)^10 * (1+4*x-x^2)^5 * (1-11*x-x^2) ).

%F 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.

%e G.f.: A(x) = 1 + x + 122*x^2 + 463*x^3 + 11985*x^4 + 85456*x^5 + ...

%e where

%e 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 + ...

%t 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 *)

%o (PARI) /* Subroutine used in PARI programs below: */

%o {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

%o (PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^5*x^k/k)+x*O(x^n)), n)}

%o (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)}

%Y Cf. A002571, A203803, A203804, A203806, A203807, A203808, A203809.

%Y Cf. A203855, A203800.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 06 2012