Real-rooted atlas

A082985 → x ↦ x+1 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-7 - 3\,x + 14\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-18 - 17\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17310
23125561
3127147567337
451176542310291810
5204737512706957165119637
681911774515769733218722471351169
73276781915864604979516830331835015271450
8131071371365453883311746131155344425474476171439629
9524287166023923058781839561926307304133647528550627197634353
1020971517340025114178051037568060851702399145641207114012127898052140483810
11838860732156323553803205647201537807860174047445602478125513319050918469101223214677493
123355443113981012526408577529860675022472212511832650044575300120606002314400304975259051250251138388161
13134217727603979767124151394615416396371288324305763754319329692752104618412243081544062630471042354511521276036619690
145368709112594876065576639123378006458527167499524472461895522996430738376074642287022944639135868425507057444750218272932010844381

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-4 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(5 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 3\,x\)
Technical details
Node id
n-8c31d57994747dda · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-481e1efad6f8d630)
Fingerprint
f6-a5f879e59b984334

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,       3;
  2:     31,      25,       5;
  3:    127,     147,      56,       7;
  4:    511,     765,     423,     102,      9;
  5:   2047,    3751,    2706,     957,    165,     11;
  6:   8191,   17745,   15769,    7332,   1872,    247,    13;
  7:  32767,   81915,   86460,   49795,  16830,   3318,   350,   15;
  8: 131071,  371365,  453883,  311746, 131155,  34442,  5474,  476,  17;
  9: 524287, 1660239, 2305878, 1839561, 926307, 304133, 64752, 8550, 627, 19;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (14*n-7+(6*n-3)*x)*P(n-1,x) - (18+17*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((14*n - 7 + (6*n - 3)*x)*P[n - 1, x] - (18 + 17*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((14*n - 7 + (6*n - 3)*x)*v[i - 1] - (18 + 17*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 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