Real-rooted atlas

A163936 → reverse → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,x^{2} + 10\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(x^{2} - x^{3} - 2\,n x^{2} + 2\,n x^{3}\right)\,P_{n-2}'(x) + \left(1 - 4\,x + 4\,n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2145
31161835
414417496315
5110498417766003465
6122843561760018540432045045
714801683012848029070020390435280675675

All 15 rows

Technical details
Node id
n-00fabd7673b77000 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-fbfce332e6b0be6b

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 0;
  1: 1;
  2: 1,    4;
  3: 1,   16,     18;
  4: 1,   44,    174,      96;
  5: 1,  104,    984,    1776,      600;
  6: 1,  228,   4356,   17600,    18540,     4320;
  7: 1,  480,  16830,  128480,   290700,   203904,    35280;
  8: 1,  988,  59850,  783200,  3220100,  4711824,  2387952,   322560;
  9: 1, 2008, 201780, 4250000, 28827500, 74640384, 77184072, 29871360, 3265920;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n^2-10*n+4)*x^2*P(n-2,x) - ((2*n-1)*x^2-(2*n-1)*x^3)*d/dx P(n-2,x) + (1+(4*n-4)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 10*n + 4)*x^2*P[n - 2, x] - ((2*n - 1)*x^2 - (2*n - 1)*x^3)*D[P[n - 2, x], x] + (1 + (4*n - 4)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 10*n + 4)*x^2*v[i - 2] - ((2*n - 1)*x^2 - (2*n - 1)*x^3)*deriv(v[i - 2], x) + (1 + (4*n - 4)*x)*v[i - 1] + (x - x^2)*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.