Real-rooted atlas

A103327 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
21121629
31248064169
4140240448256985
51605601792230410245741
6184112053761152011264409633461
7111220161344042240675845324816384195025

All 15 rows

Technical details
Node id
n-9869343d0ba9255d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-bfe4d29ca3d34feb

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: 1,   4;
  2: 1,  12,   16;
  3: 1,  24,   80,    64;
  4: 1,  40,  240,   448,    256;
  5: 1,  60,  560,  1792,   2304,    1024;
  6: 1,  84, 1120,  5376,  11520,   11264,    4096;
  7: 1, 112, 2016, 13440,  42240,   67584,   53248,   16384;
  8: 1, 144, 3360, 29568, 126720,  292864,  372736,  245760,   65536;
  9: 1, 180, 5280, 59136, 329472, 1025024, 1863680, 1966080, 1114112, 262144;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (2*n-1+4*n*x)*P(n-1,x) + (4*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[((2*n - 1 + 4*n*x)*P[n - 1, x] + (4*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] = ((2*n - 1 + 4*n*x)*v[i - 1] + (4*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

Candidate equalities ?

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