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

1,5

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.

Links

Table of n, a(n) for n=1..45.

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

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.

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/(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:

Crossrefs

Cf. A000045, A270863, A283679.

Sequence in context: A124842 A134399 A094436 * A094441 A107230 A159830

Adjacent sequences: A286009 A286010 A286011 * A286013 A286014 A286015

Keyword

nonn,tabl,more

Author

Oboifeng Dira, Apr 30 2017

Status

approved