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%I #16 Apr 15 2021 21:46:51
%S 1,1,1,-6,29,31,-283,245,298,-286,-108,119,-3106,29469,-104585,
%T -220481,3601363,-15487305,34949165,-39821950,4356011,46881744,
%U -51274736,9005908,14663472,-5205168,-1456704,-20736
%N FP3 polynomials related to the generating functions of the columns of the A156921 matrix.
%C For the matrix of the FP1 polynomials see A156921. 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... .
%C The GF3(z;m) generate the sequences of the z^m coefficients. The general structure of the GF3(z;m) is given below.
%C The FP3(z,m) in the numerator of the GF3(z;m) is a polynomial of a certain degree, let's say k3. The (k3+1) coefficients of this polynomial can be determined one by one by comparing the series expansion of the FP3(z,m) with the values of the powers of z^m in column (m+1). These values can be generated with the GF1 formulas, see A156921.
%C An appropriate name for the polynomials FP3(z;m) in the numerators of the GF(3;m) seems to be the flower polynomials of the third kind, the FP3, because the zero patterns of these polynomials look like flowers. The zero patterns of the FP3 and the FP4, see A156933, resemble each other closely and look like the zero patterns of the FP1 and FP2.
%C The sequence of the (k3+1) number of terms of the FP3(z;m) polynomials for m from 0 to 11 is 1, 2, 8, 17, 29, 45, 63, 84, 109, 137, 167, 200.
%F G.f.: GF3(z;m):= z^p*FP3(z;m)/Product_{k=0..m} (1-(k+1)*z)^(1+3*k).
%e The first few rows of the "triangle" of the FP3(z,m) coefficients are:
%e [1]
%e [1, 1]
%e [-6, 29, 31, -283, 245, 298, -286, -108]
%e The first few FP3 polynomials are:
%e FP3(z; m=0) = 1
%e FP3(z; m=1) = (1+z)
%e FP3(z; m=2) = (-6+29*z+31*z^2-283*z^3+245*z^4+298*z^5-286*z^6-108*z^7)
%e Some GF3(z;m) are:
%e GF3(z;m=1) = z^2*(1+z)/((1-z)^4*(1-2*z))
%e GF3(z;m=2) = z^2*(-6+29*z+31*z^2-283*z^3+245*z^4+298*z^5-286*z^6-108*z^7)/((1-z)^7*(1-2*z)^4*(1-3*z))
%Y Cf. A156920, A156921, A156925, A156933.
%Y For the first few GF3's see A156928, A156929, A156930, A156931.
%Y Row sums A156932.
%Y For the polynomials in the denominators of the GF3(z;m) see A157704.
%K easy,sign,tabf,uned
%O 0,4
%A _Johannes W. Meijer_, Feb 20 2009
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