0,4

For the matrix of the coefficients of the FP2 see A156925. The coefficients in the columns of this matrix are the powers of z^m, m=0, 1, 2,.. . The columns are numbered: 1, 2, 3... .

The GF4(z;m) generate the sequences of the z^m coefficients. The general structure of the GF4(z;m) is given below.

The FP4(z,m) in the nominator of the GF4(z;m) is a polynomial of a certain degree, let's say k4. The (k4+1) coefficients of this polynomial can be determined one by one by comparing the series expansion of the FP4(z,m) with the values of the powers of z^m in column (m+1). These values can be generated with the GF2 formulae, see A156925.

An appropriate name for the polynomials FP4(z;m) in the numerators of the GF4(z;m) seems to be the flower polynomials of the fourth kind because the zero patterns of these polynomials look like flowers. The zero patterns of the FP4 and the FP3, see A156927, resemble each other closely and look like the zero patterns of the FP1 and FP2.

The sequence of (k4+1) number of terms of the FP4(z;m) polynomials for m from 0 to 11 is: 1, 2, 7, 17, 28, 44, 63, 83, 108, 136, 167, 199.

G.f: GF4(z;m):= z^q*FP4(z;m) / product((1-(2*m+1-(2*k))*z)^(3*k+1), k=0..m)

The first few rows of the "triangle" of the FP4(z;m) coefficients are:

[1]

[1, 1]

[ -11, 156, -627, 736, 591, -1116, -369]

The first few FP4 polynomials are:

FP4(z; m=0) = 1

FP4(z; m=1) = (1+z)

FP4(z; m=2) = ( -11+156*z-627*z^2+736*z^3+591*z^4-1116*z^5-369*z^6 )

Some GF4(z;m) are:

GF4(z;m=1) = z*(1+z)/((1-3*z)*(1-z)^4)

GF4(z;m=2) = z^2*(-11+156*z-627*z^2+736*z^3+591*z^4-1116*z^5-369*z^6)/((1-z)^7*(1-3*z)^4*(1-5*z))

Cf. A156920, A156921, A156925, A156927.

For the first few GF4's see A156934, A156935, A156936, A156937

Row sums A156938

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 2009: (Start)

For the polynomials in the denominators of the GF4(z;m) see A157705.

(End)

Sequence in context: A157186 A122769 A051608 this_sequence A158512 A135700 A141876

Adjacent sequences: A156930 A156931 A156932 this_sequence A156934 A156935 A156936

easy,sign,tabf

Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009