Real-rooted atlas

A094816 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
315511
419231649
51146511665261
61201454706693261631
7127280141536344429195711743
813549035351408430681333751370095901

All 15 rows

Technical details
Node id
n-3b8009188e712867 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-56018d9771f15b64

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  5,    5;
   4: 1,  9,   23,    16;
   5: 1, 14,   65,   116,     65;
   6: 1, 20,  145,   470,    669,    326;
   7: 1, 27,  280,  1415,   3634,   4429,    1957;
   8: 1, 35,  490,  3535,  14084,  30681,   33375,   13700;
   9: 1, 44,  798,  7756,  43939, 147532,  284066,  283072,  109601;
  10: 1, 54, 1230, 15456, 117579, 558642, 1646714, 2878284, 2673321, 986410;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2)*x^2*P(n-2,x) + (1+n*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n - 2)*x^2*P[n - 2, x] + (1 + n*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*x^2*v[i - 2] + (1 + n*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, v[i])))};
T(10)

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