Real-rooted atlas

A123519 → basis A013610

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 11 + 42\,x + 36\,x^{2}\)
\(P_{1}(x) = 41 + 228\,x + 396\,x^{2} + 216\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
011423689
141228396216881
215311162916324012968721
35715154179643024024624777686329
42131229269986422550427864017884846656854569
579539933651904814709602442960237945612597122799368459361
629681422136257234487726241831896023996736192689288678016167961683739041
7110771176729412303144490536001234686242035212482197964161500456965878656010077696828931049
84134037311666572439962612606407698771361530900000208104422419102832641133740800393030144604661768205571449
915428412995978826054283613394529364523603760105393415681728978048019977352704159378762248364487680260004556836279705681226783441
10575796112178453211646860766659808120253612555206776808883213031412710418173781043218248188032012869217792060526642176170514616322176782336804062262961
11214890034917261065128908660322878960001368802670404127005468809105752609281488858651648180441650764816012955059201011659590656431002902528111015899136130606940167959395846169
12801980511974053910223013052001532652278407158871886402404315700928598819019788811237148361728159683360601601710298889164813593884755968777691769241630279042293767183381708807836416409678789896198729
13299303201788567784095920635600714580846560364625985456013499885388672374680797361927938887598259212949181790336016262067642163215598181278310411223997640294458662107172864210277173657604623485681664470184984576779939566141121
14111701475331364476656408695304240328058197200018156637308960734727849811202248834410789125311268339857929783321914488321410330624215040158605742490624013770738646179849050603828797444353651743293441445818827571202962165402828828211099074567720605765212481

Computed in your browser ?

Technical details
Node id
n-f5286435ed391b4f · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-02ef7371f378b284

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:      11,       42,        36;
  1:      41,      228,       396,        216;
  2:     153,     1116,      2916,       3240,       1296;
  3:     571,     5154,     17964,      30240,      24624,        7776;
  4:    2131,    22926,     99864,     225504,     278640,      178848,       46656;
  5:    7953,    99336,    519048,    1470960,    2442960,     2379456,     1259712,      279936;
  6:   29681,   422136,   2572344,    8772624,   18318960,    23996736,    19268928,     8678016,     1679616;
  7:  110771,  1767294,  12303144,   49053600,  123468624,   203521248,   219796416,   150045696,    58786560,   10077696;
  8:  413403,  7311666,  57243996,  261260640,  769877136,  1530900000,  2081044224,  1910283264,  1133740800,  393030144,   60466176;
  9: 1542841, 29959788, 260542836, 1339452936, 4523603760, 10539341568, 17289780480, 19977352704, 15937876224, 8364487680, 2600045568, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 11 + 42*x + 36*x^2, P(1,x) = 41 + 228*x + 396*x^2 + 216*x^3.
%t Mathematica
P[0, x_] := 11 + 42*x + 36*x^2;
P[1, x_] := 41 + 228*x + 396*x^2 + 216*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 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] = 11 + 42*x + 36*x^2; v[2] = 41 + 228*x + 396*x^2 + 216*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 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-8cd26e1d1ff01e6d
  • 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-8cd26e1d1ff01e6d
  • 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-8cd26e1d1ff01e6d
  • 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-8cd26e1d1ff01e6d

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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