Real-rooted atlas

A082985 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(1 + 5\,x + 5\,x^{2} + 2\,n - n x - 5\,n x^{2}\right)\,P_{n-1}(x) + \left(4\,x + 9\,x^{2} + 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
25151131
37355629127
496617118976511
5111104077156051992047
6131698322054273018725218191
71524515334965918098105655136432767

All 15 rows

Technical details
Node id
n-157478dd545a23d0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-347434f4a19cf0c5

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:  3,   4;
  2:  5,  15,   11;
  3:  7,  35,   56,    29;
  4:  9,  66,  171,   189,    76;
  5: 11, 110,  407,   715,   605,    199;
  6: 13, 169,  832,  2054,  2730,   1872,    521;
  7: 15, 245, 1533,  4965,  9180,   9810,   5655,   1364;
  8: 17, 340, 2618, 10642, 25585,  37774,  33745,  16779,  3571;
  9: 19, 456, 4218, 20862, 62453, 119168, 146357, 112309, 49096, 9349;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1-(n-5)*x-(5*n-5)*x^2)*P(n-1,x) + (4*x+9*x^2+5*x^3)*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 - (n - 5)*x - (5*n - 5)*x^2)*P[n - 1, x] + (4*x + 9*x^2 + 5*x^3)*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 - (n - 5)*x - (5*n - 5)*x^2)*v[i - 1] + (4*x + 9*x^2 + 5*x^3)*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