Real-rooted atlas

A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.

A120910 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 A120910
\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
1639
21212327
3243618381
4489672243243
596240240120303729
61925767204801803632187
73841344201616808402524236561

Show all 15 stored rows

Generated b-files: A120909 (reversed triangle), A120910 ?

Computed in your browser ?

Technical details
Node id
n-51b4f5e7901c2b50 · revision 2 · JSON
Definition source
oeis_catalog:A120910
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:    3;
  1:    6,    3;
  2:   12,   12,     3;
  3:   24,   36,    18,     3;
  4:   48,   96,    72,    24,     3;
  5:   96,  240,   240,   120,    30,    3;
  6:  192,  576,   720,   480,   180,   36,    3;
  7:  384, 1344,  2016,  1680,   840,  252,   42,   3;
  8:  768, 3072,  5376,  5376,  3360, 1344,  336,  48,  3;
  9: 1536, 6912, 13824, 16128, 12096, 6048, 2016, 432, 54, 3;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
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] = 3; 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.

A120909: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 3;
   2: 3,  6;
   3: 3, 12,  12;
   4: 3, 18,  36,   24;
   5: 3, 24,  72,   96,   48;
   6: 3, 30, 120,  240,  240,    96;
   7: 3, 36, 180,  480,  720,   576,   192;
   8: 3, 42, 252,  840, 1680,  2016,  1344,   384;
   9: 3, 48, 336, 1344, 3360,  5376,  5376,  3072,  768;
  10: 3, 54, 432, 2016, 6048, 12096, 16128, 13824, 6912, 1536;
%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) = 3.
%t Mathematica
P[1, x_] := 3;
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] = 3; 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.

A120910: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:    3;
   2:    6,    3;
   3:   12,   12,     3;
   4:   24,   36,    18,     3;
   5:   48,   96,    72,    24,     3;
   6:   96,  240,   240,   120,    30,    3;
   7:  192,  576,   720,   480,   180,   36,    3;
   8:  384, 1344,  2016,  1680,   840,  252,   42,   3;
   9:  768, 3072,  5376,  5376,  3360, 1344,  336,  48,  3;
  10: 1536, 6912, 13824, 16128, 12096, 6048, 2016, 432, 54, 3;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) for n >= 2, with P(1,x) = 3.
%t Mathematica
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 3; 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};
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, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-26e15e96a906c1df
  • 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-26e15e96a906c1df, e-3e9b67ef5f31843c, e-b63d4680864933c7
  • 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-26e15e96a906c1df, e-3e9b67ef5f31843c, e-b63d4680864933c7
  • 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-26e15e96a906c1df
  • 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-3e9b67ef5f31843c, e-b63d4680864933c7
  • 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-d8c18795222de68a
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 2] to roots in (-inf, 0]: an input root r <= 2 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-d8c18795222de68a

Proof chains

4 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 16 proof steps

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