Real-rooted atlas

A290316 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 3\,n\right)\,P_{n}(x) = \left(20 - 570\,x + 13680\,x^{2} - 12\,n + 522\,n x - 17658\,n x^{2} - 108\,n^{2} x + 7290\,n^{2} x^{2} - 972\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(15\,x - 765\,x^{2} + 2160\,x^{3} - 9\,n x + 729\,n x^{2} - 2106\,n x^{3} - 162\,n^{2} x^{2} + 486\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-37 + 624\,x + 15\,n - 522\,n x + 108\,n^{2} x\right)\,P_{n-1}(x) + \left(-24\,x + 72\,x^{2} + 9\,n x - 27\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
266
348180228
42344428648011142
599654864343764285120684744
640625214968375400270825121496880050952270
7163444357260144769032112260945622706892009160905604458531852
8654903393306020751357603138713064014578544916020541878640064126339200448826839710

All 15 rows

Technical details
Node id
n-67451616adcaaf60 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-bcf2b04cbfc6ea32

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 -2)
  • 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:       0;
   2:       6;
   3:      48,        180;
   4:     234,       4428,         6480;
   5:     996,      54864,       343764,         285120;
   6:    4062,     521496,      8375400,       27082512,        14968800;
   7:   16344,    4357260,    144769032,     1122609456,      2270689200,        916090560;
   8:   65490,   33933060,   2075135760,    31387130640,    145785449160,     205418786400,      64126339200;
   9:  262092,  253641960,  26553910860,   698551704720,   6099022863360,   19122957833904,   20103889024800,    5056819891200;
  10: 1048518, 1849403664, 315760216680, 13436013542400, 197127983736360, 1131813079627536, 2584056835942848, 2125321463174400, 443735945452800;
%F Formula
The row polynomials satisfy (3*n-8)*P(n,x) = -(12*n-20+(108*n^2-522*n+570)*x+(972*n^3-7290*n^2+17658*n-13680)*x^2)*P(n-2,x) - ((9*n-15)*x+(162*n^2-729*n+765)*x^2-(486*n^2-2106*n+2160)*x^3)*d/dx P(n-2,x) + (15*n-37+(108*n^2-522*n+624)*x)*P(n-1,x) + ((9*n-24)*x-(27*n-72)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 6.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 20 + (108*n^2 - 522*n + 570)*x + (972*n^3 - 7290*n^2 + 17658*n - 13680)*x^2)*P[n - 2, x] - ((9*n - 15)*x + (162*n^2 - 729*n + 765)*x^2 - (486*n^2 - 2106*n + 2160)*x^3)*D[P[n - 2, x], x] + (15*n - 37 + (108*n^2 - 522*n + 624)*x)*P[n - 1, x] + ((9*n - 24)*x - (27*n - 72)*x^2)*D[P[n - 1, x], x])/(3*n - 8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 6; for(n = 3, N, my(i = n); v[i] = (-(12*n - 20 + (108*n^2 - 522*n + 570)*x + (972*n^3 - 7290*n^2 + 17658*n - 13680)*x^2)*v[i - 2] - ((9*n - 15)*x + (162*n^2 - 729*n + 765)*x^2 - (486*n^2 - 2106*n + 2160)*x^3)*deriv(v[i - 2], x) + (15*n - 37 + (108*n^2 - 522*n + 624)*x)*v[i - 1] + ((9*n - 24)*x - (27*n - 72)*x^2)*deriv(v[i - 1], x))/(3*n - 8)); 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

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.