Real-rooted atlas

A123202 → x ↦ x+1

Proved for all n all roots real, in (−1, 0)coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-1-2-3
316714
4-1-12-39-34-86
5120126292209648
6-1-30-310-1360-2505-1546-5752
7142645458015675243061332758576
8-1-56-1197-12530-68635-194796-263431-130922-671568

All 15 rows

Technical details
Node id
n-184f8d63c8db0227 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ced16c5d17801fb2

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): no
  • 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,   6,     7;
   4: -1, -12,   -39,    -34;
   5:  1,  20,   126,    292,     209;
   6: -1, -30,  -310,  -1360,   -2505,    -1546;
   7:  1,  42,   645,   4580,   15675,    24306,     13327;
   8: -1, -56, -1197, -12530,  -68635,  -194796,   -263431,   -130922;
   9:  1,  72,  2044,  29624,  237510,  1068536,   2612092,   3154824,   1441729;
  10: -1, -90, -3276, -62832, -694134, -4541796, -17453100, -37705536, -41368977, -17572114;
%F Formula
The row polynomials satisfy P(n,x) = -(1+(2*n-2)*x+(n-2)*x^2)*P(n-1,x) + (x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(1 + (2*n - 2)*x + (n - 2)*x^2)*P[n - 1, x] + (x^2 + x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(1 + (2*n - 2)*x + (n - 2)*x^2)*v[i - 1] + (x^2 + x^3)*deriv(v[i - 1], x)); 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

Same fingerprint ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023