|
%I #20 Feb 26 2023 14:53:43
%S 1,3,7,19,26,81,92,267,358,848,980,3061,3030,7976,11042,25099,27150,
%T 78642,79440,219884,270704,584862,659112,1977909,1950651,4735370,
%U 6204499,14189096,14912642,43168586,41734340,110786987,135815060,290854380,339428752,953889058,893839230
%N a(n) = n*fibonacci(n) * Sum_{d|n} 1/(d*fibonacci(d)).
%H Paul D. Hanna, <a href="/A203319/b203319.txt">Table of n, a(n) for n = 1..200</a>
%F Equals the logarithmic derivative of A203318.
%F L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n satisfies:
%F (1) L(x) = Sum_{n>=1} x^n/n * Sum_{k>=0} F(n*k+n)/F(n) * x^(n*k) where F(n) = fibonacci(n).
%F (2) L(x) = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(n*k)*x^(n*k)/k ) ) where Lucas(n) = A000032(n).
%F (3) L(x) = Sum_{n>=1} x^n/n * 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) ) where Lucas(n) = A000032(n).
%F (4) L(x) = Sum_{n>=1} x^n/n * G_n(x^n) where G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-x-x^2) and u is an n-th root of unity.
%e L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 26*x^5/5 + 81*x^6/6 +...
%e where
%e L(x) = x*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 +...+ F(n+1)*x^n +...) +
%e x^2/2*(1 + 3*x^2 + 8*x^4 + 21*x^6 + 55*x^8 +...+ F(2*n+2)*x^(2*n) +...) +
%e x^3/3*(1 + 4*x^3 + 17*x^6 + 72*x^9 +...+ F(3*n+3)/2*x^(3*n) +...) +
%e x^4/4*(1 + 7*x^4 + 48*x^8 + 329*x^12 +...+ F(4*n+4)/3*x^(4*n) +...) +
%e x^5/5*(1 + 11*x^5 + 122*x^10 + 1353*x^15 +...+ F(5*n+5)/5*x^(5*n) +...) +
%e x^6/6*(1 + 18*x^6 + 323*x^12 + 5796*x^18 +...+ F(6*n+6)/8*x^(6*n) +...) +...
%e here F(n) = fibonacci(n) = A000045(n).
%e Equivalently,
%e L(x) = x/(1-x-x^2) + (x^2/2)/(1-3*x^2+x^4) + (x^3/3)/(1-4*x^3-x^6) + (x^4/4)/(1-7*x^4+x^8) +...+ (x^n/n)/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
%e here Lucas(n) = A000032(n).
%e Exponentiation of the l.g.f. equals the g.f. of A203318:
%e exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...+ A203318(n)*x^n +...
%t a[n_] := n Fibonacci[n] DivisorSum[n, 1/(# Fibonacci[#]) &]; Array[a, 40] (* _Jean-François Alcover_, Dec 23 2015 *)
%o (PARI) {a(n)=if(n<1,0, n*fibonacci(n)*sumdiv(n,d,1/(d*fibonacci(d))) )}
%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o {a(n)=n*polcoeff(sum(m=1,n+1,(x^m/m)/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o {a(n)=local(L=x); L=sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), Lucas(m*k)*x^(m*k)/k)+x*O(x^n))); n*polcoeff(L,n)}
%o (PARI) {a(n)=local(A=1+x+x*O(x^n),F=1/(1-x-x^2+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); n*polcoeff(log(A), n)}
%Y Cf. A203318, A203321; A203414 (Pell variant).
%Y Cf. A000032 (Lucas), A000045 (Fibonacci), A001906, A001076, A004187, A049666, A049660, A049667.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 01 2012
| 