Real-rooted atlas

A078812 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-8 - 24\,x - 10\,x^{2} + 8\,n + 18\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 12\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
3412824
48365021115
5169620418055551
6322406889366001442640
764576208038403804190837712649
812813445856136801844014292587898760605

All 15 rows

Technical details
Node id
n-6f364825b4b860ee · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a47082d407210578

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:   2,    3;
   3:   4,   12,     8;
   4:   8,   36,    50,     21;
   5:  16,   96,   204,    180,     55;
   6:  32,  240,   688,    936,    600,    144;
   7:  64,  576,  2080,   3840,   3804,   1908,    377;
   8: 128, 1344,  5856,  13680,  18440,  14292,   5878,    987;
   9: 256, 3072, 15680,  44352,  75840,  80064,  50828,  17700,  2584;
  10: 512, 6912, 40448, 134400, 278880, 373968, 323408, 173592, 52390, 6765;
%F Formula
The row polynomials satisfy (4*n-4)*P(n,x) = (8*n-8+(18*n-24)*x+(5*n-10)*x^2)*P(n-1,x) - (4*x+12*x^2+5*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[Together[((8*n - 8 + (18*n - 24)*x + (5*n - 10)*x^2)*P[n - 1, x] - (4*x + 12*x^2 + 5*x^3)*D[P[n - 1, x], x])/(4*n - 4)]];
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] = ((8*n - 8 + (18*n - 24)*x + (5*n - 10)*x^2)*v[i - 1] - (4*x + 12*x^2 + 5*x^3)*deriv(v[i - 1], x))/(4*n - 4)); 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. [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