Real-rooted atlas

A103327 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
24241644
389616064328
4163209608962562448
532960448071684608102418272
6642688179204300846080225284096136384
7128716864512215040337920270336106496163841017984

All 15 rows

Technical details
Node id
n-653febf5ea27da46 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-173f5abf34450e02

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:   2,     4;
  2:   4,    24,     16;
  3:   8,    96,    160,      64;
  4:  16,   320,    960,     896,      256;
  5:  32,   960,   4480,    7168,     4608,     1024;
  6:  64,  2688,  17920,   43008,    46080,    22528,     4096;
  7: 128,  7168,  64512,  215040,   337920,   270336,   106496,   16384;
  8: 256, 18432, 215040,  946176,  2027520,  2342912,  1490944,  491520,   65536;
  9: 512, 46080, 675840, 3784704, 10543104, 16400384, 14909440, 7864320, 2228224, 262144;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (4*n-2+4*n*x)*P(n-1,x) + (8*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[((4*n - 2 + 4*n*x)*P[n - 1, x] + (8*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] = ((4*n - 2 + 4*n*x)*v[i - 1] + (8*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

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