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A156921 FP1 polynomials related to the generating functions of the right hand columns of the A156920 triangle.

%I A156921 #13 Apr 15 2021 21:46:43
%S A156921 1,1,1,1,-6,1,7,-79,119,126,-270,1,28,-515,1654,8689,-65864,142371,
%T A156921 -82242,-99090,113400,1,86,-2255,5784,300930,-3904584,20663714,
%U A156921 -41517272,-80232259,657717054
%N A156921 FP1 polynomials related to the generating functions of the right hand columns of the A156920 triangle.
%C A156921 The FP1 polynomials appear in the numerators of the GF1 o.g.f.s. of the right hand columns of A156920. The FP1 can be calculated with the formula for the RHC sequence, see A156920, and the formula for the general structure of the generating function GF1, see below.
%C A156921 An appropriate name for the FP1 polynomials seems to be the flower polynomials of the first kind because the zero patterns of these polynomials look like flowers. The zero patterns of the FP2, see A156925, and the FP1 resemble each other closely.
%C A156921 A Maple program that generates for a right hand column with a certain RHCnr its GF1 and FP1 can be found below. RHCnr stands for right hand column number and starts from 1.
%F A156921 G.f.: GF1(z;RHCnr) := FP1(z;RHCnr)/product((1-(2*m-1)*z)^(RHCnr+1-m),m=1..RHCnr)
%F A156921 Row sums (n) = (-1)^(1+(n+1)*(n+2)/2)*A098695(n).
%e A156921 The first few rows of the "triangle" of the coefficients of the FP1 polynomials.
%e A156921 In the columns the coefficients of the powers of z^m, m=0,1,2,... , appear.
%e A156921   [1]
%e A156921   [1]
%e A156921   [1, 1, -6]
%e A156921   [1, 7, -79, 119, 126, -270]
%e A156921   [1, 28, -515, 1654, 8689, -65864, 142371, -82242, -99090, 113400]
%e A156921 Matrix of the coefficients of the FP1 polynomials. The coefficients in the columns of this matrix are the powers of z^m, m=0,1,2,.. .
%e A156921   [1, 0 ,0, 0, 0, 0, 0, 0, 0, 0]
%e A156921   [1, 0 ,0, 0, 0, 0, 0, 0, 0, 0]
%e A156921   [1, 1, -6, 0 ,0, 0, 0, 0, 0, 0]
%e A156921   [1, 7, -79, 119, 126, -270, 0, 0, 0, 0]
%e A156921   [1, 28, -515, 1654, 8689, -65864, 142371, -82242, -99090, 113400]
%e A156921 The first few FP1 polynomials are:
%e A156921   FP1(z; RHCnr=1) = 1
%e A156921   FP1(z; RHCnr=2) = 1
%e A156921   FP1(z; RHCnr =3) = 1+z-6*z^2
%e A156921 Some GF1(z;RHCnr) are:
%e A156921   GF1(z;RHCnr= 3) = (1+z-6*z^2)/((1-5*z)*(1-3*z)^2*(1-z)^3)
%e A156921   GF1(z;RHCnr= 4) = (1+7*z-79*z^2+119*z^3+126*z^4-270*z^5)/((1-7*z)*(1-5*z)^2*(1-3*z)^3*(1-z)^4)
%p A156921 RHCnr:=4: if RHCnr=1 then RHCmax :=1; else RHCmax:=(RHCnr-1)*(RHCnr)/2 end if: RHCend:=RHCnr+RHCmax: for k from RHCnr to RHCend do for n from 0 to k do S2[k,n]:=sum((-1)^(n+i)*binomial(n,i)*i^k/n!,i=0..n) end do: G(k,x):= sum(S2[k,p]*((2*p)!/p!) *x^p/(1-4*x)^(p+1),p=0..k)/(((-1)^(k+1)*2*x)/(-1+4*x)^(k+1)): fx:=simplify(G(k,x)): nmax:=degree(fx); RHC[k-RHCnr+1]:= coeff(fx,x,k-RHCnr)/2^(k-RHCnr) end do: a:=n-> RHC[n]: seq(a(n), n=1..RHCend-RHCnr+1); for nx from 0 to RHCmax do num:=sort(sum(A[t]*z^t, t=0..RHCmax)); nom:=Product((1-(2*u-1)*z)^(RHCnr-u+1),u=1..RHCnr): RHCa:= series(num/nom,z,nx+1); y:=coeff(RHCa,z,nx)-A[nx]; x:=RHC[nx+1]; A[nx]:=x-y; end do: FP1[RHCnr]:=sort(num,z, ascending); GenFun[RHCnr] :=FP1[RHCnr]/product((1-(2*m-1)*z)^(RHCnr-m+1),m=1..RHCnr);
%Y A156921 Cf. A156920, A156925, A156927, A156933.
%Y A156921 For the first few GF1's see A000340, A156922, A156923, A156924.
%Y A156921 The number of FP1 terms follow the triangular numbers A000217, with quite surprisingly one exception here a(0)=1.
%Y A156921 Abs(Row sums (n)) = A098695(n).
%Y A156921 For the polynomials in the denominators of the GF1(z;RHCnr) see A157702.
%K A156921 easy,sign,tabf,uned
%O A156921 0,5
%A A156921 _Johannes W. Meijer_, Feb 20 2009