A181347 Numerators of lower triangular matrix T:=log(F), with the matrix F:=A037027 (Fibonacci convolution matrix).

0, 1, 0, 1, 2, 0, -1, 2, 3, 0, -1, -1, 3, 4, 0, 2, -2, -3, 4, 5, 0, 2, 4, -1, -2, 5, 6, 0, -31, 4, 2, -4, -5, 6, 7, 0, -3, -31, 2, 8, -5, -3, 7, 8, 0, 202, -3, -31, 8, 10, -2, -7, 8, 9, 0, 4, 404, -9, -62, 2, 4, -7, -4, 9, 10, 0, -464, 8, 202, -6, -31, 4, 14, -8, -9, 10, 11, 0, -2048, -928, 12, 808, -3, -31, 14, 16, -3, -5, 11, 12, 0

Offset: 0

Author

Wolfdieter Lang, Oct 15 2010

Keywords

• sign
• easy
• tabl

Comments

The denominator triangle is given by A181348.

Because exp(T) = F, T may be considered as generator of F.

This should be read as N x N matrix for N>=2: log(F_N) := -sum(((-1)^k)/k)*(F_N - Id_N)^k,N-1) with the lower triangular N x N matrix F_N := Matrix([seq([seq(A037027(n,m),n=0..N-1)],m=0..N-1)]) and the N x N identity matrix Id_N.

The log series terminates because of the lower triangular property, and the fact that all main diagonal elements are 1, which follows from

F = Riordan matrix (Fib(x),x*Fib(x)) with the o.g.f. Fib(x)=1/(1-x-x^2). For this notation of Riordan arrays see, e.g., the W.Lang link given in A006232, and there the paragraph "Summary on A- and Z-sequences for Riordan matrices", as well as the 1991 Shapiro et al. reference on the Riordan group given in A053121.

Examples

			[0]; [1,0]; [1,2,0]; [-1,2,3,0]; [-1,-1,3,4,0];...
The rational triangle (with main diagonal elements 0) starts with the rows [0]; [1, 0]; [1, 2, 0]; [-1/2, 2, 3, 0]; [-1/3, -1, 3, 4, 0];...
		

Links

• Wolfdieter Lang, A181347/A181348, Logarithm of the Fibonacci matrix A037027

Formula

a(n,m) = numerator((log F)(n,m)), with the Fibonacci lower triangular matrix F=A037027.