Real-rooted atlas

A028326 → basis A007318

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
1426
288218
3162412254
4326448162162
56416016080202486
61283844803201202421458
7256896134411205601682824374
8512204835843584224089622432213122
91024460892161075280644032134428836239366
102048102402304030720268801612867201920360402118098
114096225285632084480844805913629568105602640440442354294
12819249152135168225280253440202752118272506881584035205284821062882
1316384106496319488585728732160658944439296219648823682288045766245223188646
143276822937674547214909442050048205004815375368785923843841281283203258247285629565938

Generated b-files: A008776 (row sums), A370469 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-416d26cbd480eee5 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-11aec07ce60b6d61

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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:    2;
  1:    4,    2;
  2:    8,    8,    2;
  3:   16,   24,   12,     2;
  4:   32,   64,   48,    16,    2;
  5:   64,  160,  160,    80,   20,    2;
  6:  128,  384,  480,   320,  120,   24,    2;
  7:  256,  896, 1344,  1120,  560,  168,   28,   2;
  8:  512, 2048, 3584,  3584, 2240,  896,  224,  32,  2;
  9: 1024, 4608, 9216, 10752, 8064, 4032, 1344, 288, 36, 2;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(2 + 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] = 2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 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.

A370469: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 2;
   2: 2,  4;
   3: 2,  8,   8;
   4: 2, 12,  24,   16;
   5: 2, 16,  48,   64,   32;
   6: 2, 20,  80,  160,  160,   64;
   7: 2, 24, 120,  320,  480,  384,   128;
   8: 2, 28, 168,  560, 1120, 1344,   896,  256;
   9: 2, 32, 224,  896, 2240, 3584,  3584, 2048,  512;
  10: 2, 36, 288, 1344, 4032, 8064, 10752, 9216, 4608, 1024;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+x)*P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-dc1b47ab878d31d1
  • 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-014dd8a0f9594ad5, e-dc1b47ab878d31d1
  • 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-014dd8a0f9594ad5, e-dc1b47ab878d31d1
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-dc1b47ab878d31d1
  • 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-014dd8a0f9594ad5

Proof chains

2 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

Same fingerprint (67)

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