Real-rooted atlas

Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.

A038330 (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 A038330
\(P_{n}(x) = \left(12 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 96\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01449616256
117281728576644096
2207362764813824307225665536
3248832414720276480921601536010241048576
4298598459719684976640221184055296073728409616777216
53583180883607552836075524644864015482880309657634406416384268435456
64299816961146617856133772083289181388837158912099090432165150721572864655364294967296
751597803521547934105620639121408160526499848026324992267544166459454259284934656707788826214468719476736

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-a93042cd70c4761e · revision 1 · JSON
Definition source
oeis_catalog:A038330
Fingerprint
f6-f9762973ec1089d7

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:          144,            96,            16;
  1:         1728,          1728,           576,            64;
  2:        20736,         27648,         13824,          3072,           256;
  3:       248832,        414720,        276480,         92160,         15360,          1024;
  4:      2985984,       5971968,       4976640,       2211840,        552960,         73728,         4096;
  5:     35831808,      83607552,      83607552,      46448640,      15482880,       3096576,       344064,        16384;
  6:    429981696,    1146617856,    1337720832,     891813888,     371589120,      99090432,     16515072,      1572864,       65536;
  7:   5159780352,   15479341056,   20639121408,   16052649984,    8026324992,    2675441664,    594542592,     84934656,     7077888,     262144;
  8:  61917364224,  206391214080,  309586821120,  275188285440,  160526499840,   64210599936,  17836277760,   3397386240,   424673280,   31457280,   1048576;
  9: 743008370688, 2724364025856, 4540606709760, 4540606709760, 3027071139840, 1412633198592, 470877732864, 112113745920, 18685624320, 2076180480, 138412032, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = (12+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 96*x + 16*x^2.
%t Mathematica
P[0, x_] := 144 + 96*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 4*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] = 144 + 96*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 4*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-016885d52a801213
  • 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-016885d52a801213, e-0a35d6a32dc0856a
  • 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-016885d52a801213, e-0a35d6a32dc0856a
  • 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-016885d52a801213
  • 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-0a35d6a32dc0856a
  • 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-19cc8e386ff5c847
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 12] to roots in (-inf, 0]: an input root r <= 12 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-19cc8e386ff5c847

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 (16)

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