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A286012
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A Kedlaya-Wilf matrix for the Fibonacci sequence A000045.
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0
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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
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1,5
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COMMENTS
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For any power series f(x) starting with the term x the first column of the Kedlaya-Wilf 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.
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LINKS
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FORMULA
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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 j-fold composition of f with itself.
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EXAMPLE
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f^(3)(x) = x + 3x^2 + 12x^4 + ... as in A283679, so a(4)=1, a(8)=3, a(13)=12.
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MAPLE
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h:= x-> x/(1-x-x^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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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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