Real-rooted atlas

A086645 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
318817
4118483299
5132160256128577
6150400112012805123363
7172840358469126144204819601
8198156894082688039424286728192114243

All 15 rows

Technical details
Node id
n-a69c728081f82ab6 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-607690c148a1ebbd

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,   8,    8;
   4: 1,  18,   48,    32;
   5: 1,  32,  160,   256,    128;
   6: 1,  50,  400,  1120,   1280,    512;
   7: 1,  72,  840,  3584,   6912,   6144,    2048;
   8: 1,  98, 1568,  9408,  26880,  39424,   28672,    8192;
   9: 1, 128, 2688, 21504,  84480, 180224,  212992,  131072,  32768;
  10: 1, 162, 4320, 44352, 228096, 658944, 1118208, 1105920, 589824, 131072;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2+(2*n-4)*x)*P(n-1,x) + (2*x+2*x^2)*d/dx 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[Together[((n - 2 + (2*n - 4)*x)*P[n - 1, x] + (2*x + 2*x^2)*D[P[n - 1, x], x])/(n - 2)]];
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 + (2*n - 4)*x)*v[i - 1] + (2*x + 2*x^2)*deriv(v[i - 1], x))/(n - 2)); 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