Real-rooted atlas

Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).

A123519 (tail) nonnegative coefficients

Proved for all n: all roots real, in (−2, 0) · interlacing · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A123519
\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1 + 6\,x + 4\,x^{2}\)
\(P_{1}(x) = 1 + 12\,x + 20\,x^{2} + 8\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
016411
111220841
2120605616153
313014022414432571
4142280672720352642131
51565041680264021128321287953
61728403696792091525824192025629681
719013207392205923203229120153604352512110771

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-8eb9bb3a09f0b7fd · revision 1 · JSON
Definition source
oeis_catalog:A123519
Fingerprint
f6-173f5abf34450e02

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): no

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: 1,   6,    4;
  1: 1,  12,   20,     8;
  2: 1,  20,   60,    56,     16;
  3: 1,  30,  140,   224,    144,     32;
  4: 1,  42,  280,   672,    720,    352,     64;
  5: 1,  56,  504,  1680,   2640,   2112,    832,    128;
  6: 1,  72,  840,  3696,   7920,   9152,   5824,   1920,    256;
  7: 1,  90, 1320,  7392,  20592,  32032,  29120,  15360,   4352,   512;
  8: 1, 110, 1980, 13728,  48048,  96096, 116480,  87040,  39168,  9728,  1024;
  9: 1, 132, 2860, 24024, 102960, 256256, 396032, 391680, 248064, 97280, 21504, 2048;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1 + 6*x + 4*x^2, P(1,x) = 1 + 12*x + 20*x^2 + 8*x^3.
%t Mathematica
P[0, x_] := 1 + 6*x + 4*x^2;
P[1, x_] := 1 + 12*x + 20*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 6*x + 4*x^2; v[2] = 1 + 12*x + 20*x^2 + 8*x^3; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - v[i - 2]); 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, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-b5382d7d87df09a0
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-2a4594def0262eef, e-8d057b494ddd47e8, e-b5382d7d87df09a0
  • 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-8d057b494ddd47e8
  • 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-2a4594def0262eef, e-8d057b494ddd47e8
  • 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-2a4594def0262eef, e-8d057b494ddd47e8
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-2a4594def0262eef

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 16 proof steps

Same fingerprint (2)

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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023