Real-rooted atlas

Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.

A038226 (tail) nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A038226
\(P_{n}(x) = \left(3 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 9 + 48\,x + 64\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
094864121
1272165765121331
28186434566144409614641
3243324017280460806144032768161051
472911664777602764805529605898242621441771561
52187408243265921451520387072061931525505024209715219487171
66561139968130636869672962322432049545216660602885033164816777216214358881
7196834723925038848313528321254113283344302085945425926794772484529848321342177282357947691

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-26633b980c3508e8 · revision 1 · JSON
Definition source
oeis_catalog:A038226
Fingerprint
f6-10cba46fcb583bff

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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 ?

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:      9,      48,       64;
  1:     27,     216,      576,       512;
  2:     81,     864,     3456,      6144,       4096;
  3:    243,    3240,    17280,     46080,      61440,       32768;
  4:    729,   11664,    77760,    276480,     552960,      589824,      262144;
  5:   2187,   40824,   326592,   1451520,    3870720,     6193152,     5505024,     2097152;
  6:   6561,  139968,  1306368,   6967296,   23224320,    49545216,    66060288,    50331648,    16777216;
  7:  19683,  472392,  5038848,  31352832,  125411328,   334430208,   594542592,   679477248,   452984832,   134217728;
  8:  59049, 1574640, 18895680, 134369280,  627056640,  2006581248,  4459069440,  6794772480,  6794772480,  4026531840,  1073741824;
  9: 177147, 5196312, 69284160, 554273280, 2956124160, 11036196864, 29429858304, 56056872960, 74742497280, 66437775360, 35433480192, 8589934592;
%F Formula
The row polynomials satisfy P(n,x) = (3+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 48*x + 64*x^2.
%t Mathematica
P[0, x_] := 9 + 48*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 9 + 48*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 8*x)*v[i - 1]); 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-fdf6b848b46b3343
  • 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-711ca5eff5ba56ea, e-fdf6b848b46b3343
  • 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-711ca5eff5ba56ea, e-fdf6b848b46b3343
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-fdf6b848b46b3343
  • 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-711ca5eff5ba56ea
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-71cb1ba62d8762b5
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 3] to roots in (-inf, 0]: an input root r <= 3 gives the root (r - a)/d <= 0; a larger root gives a positive root
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-71cb1ba62d8762b5

Proof chains

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

Used in 7 proof steps

Same fingerprint (4)

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

References

  1. [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