

A286012


A KedlayaWilf matrix for the Fibonacci sequence A000045.


0



1, 1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 4, 12, 17, 5, 1, 5, 20, 48, 50, 8, 1, 6, 30, 102, 197, 147, 13, 1, 7, 42, 185, 532, 815, 434, 21, 1, 8, 56, 303, 1165, 2804, 3391, 1282, 34
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OFFSET

1,5


COMMENTS

For any power series f(x) starting with the term x the first column of the KedlayaWilf matrix are the coefficients of f(x), the second column are the coefficients of f(f(x)), the third column are the coefficients of f(f(f(x))) and so on. This gives a matrix with first row consisting of ones. The sequence given is the diagonal reading of this matrix from right up to left down.


LINKS



FORMULA

As an n X n matrix a(i,j) = coefficient of x^i in f^(j)(x) for i,j=1..n where f^(j) is the jfold composition of f with itself.


EXAMPLE

f^(3)(x) = x + 3x^2 + 12x^4 + ... as in A283679, so a(4)=1, a(8)=3, a(13)=12.


MAPLE

h:= x> x/(1xx^2):
h2:= n> coeff(series(h(h(x))), x, n+1), x, n):
h3:= n > coeff(series(h(h2(x))), x, n+1), x, n):
etc.
h7:= n > coeff(series(h(h6(x))), x, n+1), x, n): N7:=array(1..7, 1..7, sparse): gg:=array([h1, h2, h3, h4, h5, h6, h7]):for k from 1 to 7 do: for j from 1 to 7 do: N7[k, j]:=coeff(series(gg[j], x, 12), x^k): od:od:


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KEYWORD



AUTHOR



STATUS

approved



