Real-rooted atlas

Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.

A038284 (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 A038284
\(P_{n}(x) = \left(8 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 64 + 96\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0649636196
151211528642162744
2409612288138246912129638416
332768122880184320138240518407776537824
42621441179648221184022118401244160373248466567529536
5209715211010048247726083096576023224320104509442612736279936105413504
616777216100663296264241152396361728371589120222953472836075521791590416796161475789056
7134217728905969664271790899247563407365350883328401316249620065812486449725441209323521007769620661046784

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-c11cb6bca0e3e7dc · revision 1 · JSON
Definition source
oeis_catalog:A038284
Fingerprint
f6-c91d5d506c107735

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:         64,          96,           36;
  1:        512,        1152,          864,          216;
  2:       4096,       12288,        13824,         6912,         1296;
  3:      32768,      122880,       184320,       138240,        51840,         7776;
  4:     262144,     1179648,      2211840,      2211840,      1244160,       373248,        46656;
  5:    2097152,    11010048,     24772608,     30965760,     23224320,     10450944,      2612736,       279936;
  6:   16777216,   100663296,    264241152,    396361728,    371589120,    222953472,     83607552,     17915904,      1679616;
  7:  134217728,   905969664,   2717908992,   4756340736,   5350883328,   4013162496,   2006581248,    644972544,    120932352,    10077696;
  8: 1073741824,  8053063680,  27179089920,  54358179840,  71345111040,  64210599936,  40131624960,  17199267840,   4837294080,   806215680,   60466176;
  9: 8589934592, 70866960384, 265751101440, 597939978240, 896909967360, 941755465728, 706316599296, 378383892480, 141893959680, 35473489920, 5321023488, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = (8+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 96*x + 36*x^2.
%t Mathematica
P[0, x_] := 64 + 96*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 6*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] = 64 + 96*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (8 + 6*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-4ef268212a4f2f46
  • 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-4ef268212a4f2f46, e-4fdd1b3030c1c25c
  • 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-4ef268212a4f2f46, e-4fdd1b3030c1c25c
  • 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-4ef268212a4f2f46
  • 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-4fdd1b3030c1c25c
  • 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-39fc4711b1b8289f
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 8] to roots in (-inf, 0]: an input root r <= 8 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-39fc4711b1b8289f

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 (6)

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