Real-rooted atlas

A172431 → basis A013610 → basis A008277

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-16 - 33\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(-33\,x - 18\,x^{2}\right)\,P_{n-2}'(x) + \left(-9\,x^{2}\right)\,P_{n-2}''(x) + \left(9 + 6\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19615
26511127203
344114648461082859
429291702217604513040543090
519305186615308583154332271351458697428
61268811984320494505037836181113021130977510312088970
783304920744886752025608277411636280710702172859194817496223466493
85467345214752831110687582716876177201035294534293329188401708162545668590494386112978
93587732122101065041594716672632842427184271243930501056232013420909600822132587441053040519683091037236980
102354183692266355038222646195708461891393210267033950896734489052110892627308620135268661881067192577422200356495391990769124971
11154472818523188659565531839274008631139883225253215890027102181105161266937343682262420928722461059540810085554635090653656164982906212576445713334571407
12101358597612369284529040444496394488502064457337060703652733038849833050077517182711355123620550973432033198921651311575367914555541776502333923534863099761469087331098999232047424
1366507086889241866638769666174252765817203692552026058640820449273970447285195780678827004709393963810928149802879997364428687559379417933754076396469243125760348910349282034023696954192232052227588890857606377
144363900258252467695753807518545162112874027654257786840157601810327811845760282311709942040784561566488003478088326135331227302829614640710092484706219287493910602020828759448439812414426684894446034711213876080488571744535721478776691808626
Technical details
Node id
n-35e2dc42853b99b2 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-54d16d35b611849b

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        9,          6;
  2:       65,        111,          27;
  3:      441,       1464,         846,         108;
  4:     2929,      17022,       17604,        5130,         405;
  5:    19305,     186615,      308583,      154332,       27135,        1458;
  6:   126881,    1984320,     4945050,     3783618,     1113021,      130977,       5103;
  7:   833049,   20744886,    75202560,    82774116,    36280710,     7021728,     591948,     17496;
  8:  5467345,  214752831,  1106875827,  1687617720,  1035294534,   293329188,   40170816,   2545668,    59049;
  9: 35877321, 2210106504, 15947166726, 32842427184, 27124393050, 10562320134, 2090960082, 213258744, 10530405, 196830;
%F Formula
The row polynomials satisfy P(n,x) = -(16+33*x+9*x^2)*P(n-2,x) - (33*x+18*x^2)*d/dx P(n-2,x) - 9*x^2*d^2/dx^2 P(n-2,x) + (9+6*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(16 + 33*x + 9*x^2)*P[n - 2, x] - (33*x + 18*x^2)*D[P[n - 2, x], x] - 9*x^2*D[P[n - 2, x], {x, 2}] + (9 + 6*x)*P[n - 1, x] + 6*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(16 + 33*x + 9*x^2)*v[i - 2] - (33*x + 18*x^2)*deriv(v[i - 2], x) - 9*x^2*deriv(deriv(v[i - 2], x), x) + (9 + 6*x)*v[i - 1] + 6*x*deriv(v[i - 1], x)); 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)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023