A158258 L.g.f.: exp(Sum_{n>=1} a(n)*x^n/n) = 1 + x*exp(Sum_{n>=1} Lucas(n)*a(n)*x^n/n) where Lucas(n) = A000204(n).

1, 1, 4, 21, 186, 2482, 52431, 1742069, 92198200, 7788221136, 1053871857226, 228795949744458, 79812945269217967, 44781474458725910347, 40447360752560508229164, 58848264986153917140728453

Offset: 0

Author

Paul D. Hanna, Mar 28 2009

Keywords

• nonn

Examples

			L.g.f.: A(x) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 186*x^5/5 + 2482*x^6/6 +...
exp(A(x)) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 44*x^5 + 458*x^6 + 7953*x^7 +...
exp(A(x)) = 1 + x*G(x) where G(x) is the g.f. of A158257 such that:
log(G(x)) = x + 3*1*x^2/2 + 4*4*x^3/3 + 7*21*x^4/4 + 11*186*x^5/5 + 18*2482*x^6/6 +...
		

Crossrefs

Cf. A158257, A158108 (variant), A000204 (Lucas).

Programs

PARI

{a(n)=local(A=x+x^2);if(n==0,1,for(i=1,n-1,A=log(1+x*exp(sum(m=1,n,(fibonacci(m-1)+fibonacci(m+1))*x^m*polcoeff(A+x*O(x^m),m) )+x*O(x^n))));n*polcoeff(A,n))}

Formula

L.g.f.: exp(Sum_{n>=1} a(n)*x^n/n) = 1 + x*G(x) where G(x) = g.f. of A158257.

exp(Sum_{n>=1} a(n)*x^n/n) = (1 + Sum_{n>=1} Lucas(n)*a(n)*x^n) / (1 + Sum_{n>=1} (Lucas(n)-1)*a(n)*x^n).