The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A156921 FP1 polynomials related to the generating functions of the right hand columns of the A156920 triangle. 14
 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 (list; graph; refs; listen; history; text; internal format)
 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 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, -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,changed AUTHOR Johannes W. Meijer, Feb 20 2009 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 20 23:46 EDT 2021. Contains 343143 sequences. (Running on oeis4.)