Real-rooted atlas

A116414 → basis A013609 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3\right)\,P_{n-2}(x) + \left(7 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17815
24611264222
3301112813445123285
4196999681824014336409648609
5128808214420339224473614336032768719280
684253648144202617632972802949120137625626214410643373
7551131496460018758208385561604659200033128448128450562097152157492755
8360515837216816164945728410066944625745920600506368354156544117440512167772162330461206
923582713274465176139608000040743659527520980992911012659272446115843649044480105696460813421772834484439825
101542635171998267488114734442243845900083283364556800122137214976122530824192831478824963654077644893952409601073741824510275213757
11100909648014398585024921120094723487774617608686609612801494546579456181307965440015513346375689177976012803576902451208267812044885899345927550674886880
12660088480910286806454472555241382430629613056008620752752640170447021015042428033774387225114536050688187256379801609818026803200343704757862472155450572868719476736111729297661929
1343178904223729687775208562547538470426198816064512822429768294401837952969932803008807420559363653904503930883292423614627842174582207283201023555131146243252149236531262534723829765497558138881653287440268295
14282449675134514464146660843039172653120219206631612416759429108064256189337678728396835036865571061764889435481047040517164321944371241266923563909122445843214893056104432988795699230374008717312053876069761024439804651110424464123711038638
Technical details
Node id
n-23ea183a38c5e3bb · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-62f0312d0cee3fb2

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:        7,         8;
  2:       46,       112,         64;
  3:      301,      1128,       1344,        512;
  4:     1969,      9968,      18240,      14336,       4096;
  5:    12880,     82144,     203392,     244736,     143360,      32768;
  6:    84253,    648144,    2026176,    3297280,    2949120,    1376256,     262144;
  7:   551131,   4964600,   18758208,   38556160,   46592000,   33128448,   12845056,    2097152;
  8:  3605158,  37216816,  164945728,  410066944,  625745920,  600506368,  354156544,  117440512,   16777216;
  9: 23582713, 274465176, 1396080000, 4074365952, 7520980992, 9110126592, 7244611584, 3649044480, 1056964608, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = -3*P(n-2,x) + (7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 8*x;
P[n_, x_] := P[n, x] = Expand[-3*P[n - 2, x] + (7 + 8*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] = 1; v[2] = 7 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -3*v[i - 2] + (7 + 8*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)

polytool c48629374597

Properties

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