Real-rooted atlas

A111125 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-12 - 9\,x + 8\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-5 - 9\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2437
31115531
42956357127
576189171669511
6199605715407110112047
7521187227302054832169138191
81364565598109180496515332451532767
935711677933745377742558510642261834017131071
109349490961123091463571191686245320862421845619524287
1124476142065364245541646515340326109138005381606489595212097151
12640794073531157015193335721051211561861801297282095660339614759238388607
131677611159200361290067019008219200699739042220501808600541475109175138059502533554431
144392043277611111213002267676030932550297251102063079010423350381010598670017374519305117027134217727
151149851921651933820061751859801129152701209027409490656055281540238962907579585172100526767026390142129536870911
Technical details
Node id
n-51b406bd20106615 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-347434f4a19cf0c5

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    4,     3;
   3:   11,    15,      5;
   4:   29,    56,     35,      7;
   5:   76,   189,    171,     66,      9;
   6:  199,   605,    715,    407,    110,    11;
   7:  521,  1872,   2730,   2054,    832,   169,    13;
   8: 1364,  5655,   9810,   9180,   4965,  1533,   245,   15;
   9: 3571, 16779,  33745,  37774,  25585, 10642,  2618,  340,  17;
  10: 9349, 49096, 112309, 146357, 119168, 62453, 20862, 4218, 456, 19;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = (8*n-12+(6*n-9)*x)*P(n-1,x) - (5+9*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 12 + (6*n - 9)*x)*P[n - 1, x] - (5 + 9*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 12 + (6*n - 9)*x)*v[i - 1] - (5 + 9*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023