Real-rooted atlas

A091042 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
26410
320401676
45622422464568
5144960172811522564240
635235209856112645632102431648
783211648465927987266560266244096236224
8192035840193536460800563200368640122880163841763200
9435210444873113622978563829760362086419496965570566553613160704
1097282918402568192102727682225766428327936217907209961472249036826214498232832
112150478848085155844217241611480268818785894419267584012478054449545216110100481048576733219840
12471042072576269434881616609285388697601097334784143497625612299796486873415682411724804823449641943045472827392
13102400532480082001920585728000234291200057933824009358540800101607014407471104000367001600011534336002097152001677721640849739776
142211841341849624153292820242759689559080960281558384645486778777673157050368677778554884359979008019100860416543581798490596966467108864304906608640
1547513633259520691798016672032358436961779712127686148096294660341760471456546816533855207424430830518272246188867584973078528002530004172838923141122684354562275853910016
Technical details
Node id
n-e075816292b6a244 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-3e49ca6e8ed7236e

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:    6,      4;
   3:   20,     40,      16;
   4:   56,    224,     224,       64;
   5:  144,    960,    1728,     1152,      256;
   6:  352,   3520,    9856,    11264,     5632,     1024;
   7:  832,  11648,   46592,    79872,    66560,    26624,     4096;
   8: 1920,  35840,  193536,   460800,   563200,   368640,   122880,   16384;
   9: 4352, 104448,  731136,  2297856,  3829760,  3620864,  1949696,  557056,   65536;
  10: 9728, 291840, 2568192, 10272768, 22257664, 28327936, 21790720, 9961472, 2490368, 262144;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = (4*n-2+(4*n-4)*x)*P(n-1,x) + (8*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[((4*n - 2 + (4*n - 4)*x)*P[n - 1, x] + (8*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] = ((4*n - 2 + (4*n - 4)*x)*v[i - 1] + (8*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