A286012 A Kedlaya-Wilf matrix for the Fibonacci sequence A000045.

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

Offset: 1

Author

Oboifeng Dira, Apr 30 2017

Keywords

• nonn
• tabl
• more

Comments

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.

Examples

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

Links

• Kiran S. Kedlaya, Another Combinatorial Determinant, Journal of Combinatorial Theory Series A 90(1), November 1998.

Crossrefs

Cf. A000045, A270863, A283679.

Programs

Maple

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:

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 j-fold composition of f with itself.