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

1, 1, 1, 1, -6, 1, 7, -79, 119, 126, -270, 1, 28, -515, 1654, 8689, -65864, 142371, -82242, -99090, 113400, 1, 86, -2255, 5784, 300930, -3904584, 20663714, -41517272, -80232259, 657717054

Offset

0,5

Comments

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.

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.

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.

Links

Table of n, a(n) for n=0..30.

Formula

G.f.: GF1(z;RHCnr) := FP1(z;RHCnr)/product((1-(2*m-1)*z)^(RHCnr+1-m),m=1..RHCnr)

Row sums (n) = (-1)^(1+(n+1)*(n+2)/2)*A098695(n).

Example

The first few rows of the "triangle" of the coefficients of the FP1 polynomials.
In the columns the coefficients of the powers of z^m, m=0,1,2,... , appear.
  [1]
  [1]
  [1, 1, -6]
  [1, 7, -79, 119, 126, -270]
  [1, 28, -515, 1654, 8689, -65864, 142371, -82242, -99090, 113400]
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,.. .
  [1, 0 ,0, 0, 0, 0, 0, 0, 0, 0]
  [1, 0 ,0, 0, 0, 0, 0, 0, 0, 0]
  [1, 1, -6, 0 ,0, 0, 0, 0, 0, 0]
  [1, 7, -79, 119, 126, -270, 0, 0, 0, 0]
  [1, 28, -515, 1654, 8689, -65864, 142371, -82242, -99090, 113400]
The first few FP1 polynomials are:
  FP1(z; RHCnr=1) = 1
  FP1(z; RHCnr=2) = 1
  FP1(z; RHCnr =3) = 1+z-6*z^2
Some GF1(z;RHCnr) are:
  GF1(z;RHCnr= 3) = (1+z-6*z^2)/((1-5*z)*(1-3*z)^2*(1-z)^3)
  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)

Maple

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);

Crossrefs

Cf. A156920, A156925, A156927, A156933.

For the first few GF1's see A000340, A156922, A156923, A156924.

The number of FP1 terms follow the triangular numbers A000217, with quite surprisingly one exception here a(0)=1.

Abs(Row sums (n)) = A098695(n).

For the polynomials in the denominators of the GF1(z;RHCnr) see A157702.

Sequence in context: A204205 A143019 A337369 * A094214 A001622 A186099

Adjacent sequences: A156918 A156919 A156920 * A156922 A156923 A156924

Keyword

easy,sign,tabf,uned

Author

Johannes W. Meijer, Feb 20 2009

Status

approved