Real-rooted atlas

A007318 → ×(1+2n+n²) → θ+1

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(n^{2} - 2\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(-n^{2} - 2\,n^{2} x - n^{2} x^{2} - 2\,n^{3} - 4\,n^{3} x - 2\,n^{3} x^{2} - n^{4} - 2\,n^{4} x - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(2 + 2\,x - 4\,n^{2} - 4\,n^{2} x + 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
29362772
3169614464320
4252004504001251200
536360108014409002164032
649588220539203675176434312544
76489640328960112008064313651236864

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-a9e082eabd70d305 · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e99c3824c4864868

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   4,    8;
  2:   9,   36,    27;
  3:  16,   96,   144,    64;
  4:  25,  200,   450,   400,   125;
  5:  36,  360,  1080,  1440,   900,   216;
  6:  49,  588,  2205,  3920,  3675,  1764,   343;
  7:  64,  896,  4032,  8960, 11200,  8064,  3136,   512;
  8:  81, 1296,  6804, 18144, 28350, 27216, 15876,  5184,  729;
  9: 100, 1800, 10800, 33600, 63000, 75600, 58800, 28800, 8100, 1000;
%F Formula
The row polynomials satisfy (n^4-2*n^3+n^2)*P(n,x) = -(n^4+2*n^3+n^2+(2*n^4+4*n^3+2*n^2)*x+(n^4+2*n^3+n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^2+2+(2*n^4-4*n^2+2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + 2*n^3 + n^2 + (2*n^4 + 4*n^3 + 2*n^2)*x + (n^4 + 2*n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^2 + 2 + (2*n^4 - 4*n^2 + 2)*x)*P[n - 1, x])/(n^4 - 2*n^3 + n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + 2*n^3 + n^2 + (2*n^4 + 4*n^3 + 2*n^2)*x + (n^4 + 2*n^3 + n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^2 + 2 + (2*n^4 - 4*n^2 + 2)*x)*v[i - 1])/(n^4 - 2*n^3 + n^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-d99983198f1cc02c
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-d99983198f1cc02c
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-d99983198f1cc02c

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (27)

Equal up to content, sign, reversal and a small index shift.

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  5. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  6. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.