Real-rooted atlas

A110441 → basis A007318 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2\right)\,P_{n-2}(x) + \left(5 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
22320447
3105142608315
4479880576160162111
521855074452019364003214147
6996727980315961840058409606494807
74546514968620490015132065200164162240128635355
8207391783400126068011296006169602105604390451202564257871
9946025403241074604007866720521360022538886361601131521152051228534387
104315343204873004284546051995200405675202127552076007681827840283392256001024191224967
1119684665103002366240281100329933480296400800183004864792825602411443250496006937605632020481281505995
128979263951340656013217193122026239200206073592014652748807472209922754816007291008013516800166707212288040968588092031
134095938652540613346715484748012114767664137633564001108183651265080896002823621120905414400212016640352563203944448266240819257553632227
1418683840471249544134038212025468708310548808892484548880005345600532096790402658332160010028494080284387840059698073689989120921190457344016384385699241527
Technical details
Node id
n-2f962364d2437cec · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-63895f857cd2fa4e

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:      5,       2;
  2:     23,      20,       4;
  3:    105,     142,      60,       8;
  4:    479,     880,     576,     160,      16;
  5:   2185,    5074,    4520,    1936,     400,      32;
  6:   9967,   27980,   31596,   18400,    5840,     960,     64;
  7:  45465,  149686,  204900,  151320,   65200,   16416,   2240,    128;
  8: 207391,  783400, 1260680, 1129600,  616960,  210560,  43904,   5120,   256;
  9: 946025, 4032410, 7460400, 7866720, 5213600, 2253888, 636160, 113152, 11520, 512;
%F Formula
The row polynomials satisfy P(n,x) = -2*P(n-2,x) + (5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-2*P[n - 2, x] + (5 + 2*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] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -2*v[i - 2] + (5 + 2*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