Real-rooted atlas

A114687 → basis A013609

nonnegative coefficients

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

Recurrence
\(\left(2 + n\right)\,P_{n}(x) = \left(1 + 8\,x + 16\,x^{2} - n - 8\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(3 + 4\,x + 6\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
211281655
34516819264469
4197968166411522564237
5903550012640134406400102439907
64279310928952012896096000337924096387739
72079317550460748811065601128960630784172032163843858505

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-146d64c3583c8509 · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f60b2631b48d7d2e

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

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;
  1:      3,       4;
  2:     11,      28,       16;
  3:     45,     168,      192,       64;
  4:    197,     968,     1664,     1152,       256;
  5:    903,    5500,    12640,    13440,      6400,      1024;
  6:   4279,   31092,    89520,   128960,     96000,     33792,     4096;
  7:  20793,  175504,   607488,  1106560,   1128960,    630784,   172032,    16384;
  8: 103049,  990736,  4006912,  8834560,  11490304,   8888320,  3899392,   851968,   65536;
  9: 518859, 5597172, 25903296, 67081728, 106251264, 105541632, 64684032, 23003136, 4128768, 262144;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(n-1+(8*n-8)*x+(16*n-16)*x^2)*P(n-2,x) + (6*n+3+(8*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (8*n - 8)*x + (16*n - 16)*x^2)*P[n - 2, x] + (6*n + 3 + (8*n + 4)*x)*P[n - 1, x])/(n + 2)]];
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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (8*n - 8)*x + (16*n - 16)*x^2)*v[i - 2] + (6*n + 3 + (8*n + 4)*x)*v[i - 1])/(n + 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, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-01612a21c966f541, e-452beb3fe1708cbf
  • 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-01612a21c966f541, e-452beb3fe1708cbf
  • 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-01612a21c966f541, e-452beb3fe1708cbf
  • 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-01612a21c966f541, e-452beb3fe1708cbf

Proof chains

2 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

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