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

%I A181347 #8 Aug 13 2019 11:05:45
%S A181347 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,
%T A181347 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,
%U A181347 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
%N A181347 Numerators of lower triangular matrix T:=log(F), with the matrix F:=A037027 (Fibonacci convolution matrix).
%C A181347 The denominator triangle is given by A181348.
%C A181347 Because exp(T) = F, T may be considered as generator of F.
%C A181347 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.
%C A181347 The log series terminates because of the lower triangular property, and the fact that all main diagonal elements are 1, which follows from
%C A181347 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.
%H A181347 Wolfdieter Lang, <a href="/A181347/a181347.pdf">A181347/A181348, Logarithm of the Fibonacci matrix A037027</a>
%F A181347 a(n,m) = numerator((log F)(n,m)), with the Fibonacci lower triangular matrix F=A037027.
%e A181347 [0]; [1,0]; [1,2,0]; [-1,2,3,0]; [-1,-1,3,4,0];...
%e A181347 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];...
%K A181347 sign,easy,tabl
%O A181347 0,5
%A A181347 _Wolfdieter Lang_, Oct 15 2010