Real-rooted atlas

A257624 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + 3\,n\right)\,P_{n}(x) = \left(-80 + 80\,x - 8\,x^{2} - 120\,n + 60\,n x + 18\,n x^{2} - 90\,n^{2} x + 27\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-30\,x + 36\,x^{2} - 6\,x^{3} - 45\,n x + 36\,n x^{2} + 9\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 2\,x + 39\,n - 12\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x + 3\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
155
28050130
391518303753120
490704026027210250079040
5831856968201045230332740156252173600
672798010493310292664402098662036399009375065208000
76183215144545610677309565903079420361364025370992905468752130128000

All 15 rows

Technical details
Node id
n-9dfac173d10f6d59 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-62f862e74c2429e2

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:         5;
  2:        80,          50;
  3:       915,        1830,          375;
  4:      9070,       40260,        27210,         2500;
  5:     83185,      696820,      1045230,       332740,         15625;
  6:    727980,    10493310,     29266440,     20986620,       3639900,        93750;
  7:   6183215,   144545610,    677309565,    903079420,     361364025,     37099290,      546875;
  8:  51496970,  1874429600,  13818702210,  30704700280,   23031170350,   5623288800,   360478790,    3125000;
  9: 423303885, 23296568040, 258129652140, 890478630360, 1113098287950, 516259304280, 81537988140, 3386431080, 17578125;
%F Formula
The row polynomials satisfy (3*n-1)*P(n,x) = -(120*n+80+(90*n^2-60*n-80)*x+(27*n^3-27*n^2-18*n+8)*x^2)*P(n-2,x) - ((45*n+30)*x+(27*n^2-36*n-36)*x^2-(27*n^2+9*n-6)*x^3)*d/dx P(n-2,x) + (39*n+2+(18*n^2-12*n+2)*x)*P(n-1,x) + ((9*n-3)*x-(9*n-3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[(-(120*n + 80 + (90*n^2 - 60*n - 80)*x + (27*n^3 - 27*n^2 - 18*n + 8)*x^2)*P[n - 2, x] - ((45*n + 30)*x + (27*n^2 - 36*n - 36)*x^2 - (27*n^2 + 9*n - 6)*x^3)*D[P[n - 2, x], x] + (39*n + 2 + (18*n^2 - 12*n + 2)*x)*P[n - 1, x] + ((9*n - 3)*x - (9*n - 3)*x^2)*D[P[n - 1, x], x])/(3*n - 1)]];
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] = 5; for(n = 2, N, my(i = n + 1); v[i] = (-(120*n + 80 + (90*n^2 - 60*n - 80)*x + (27*n^3 - 27*n^2 - 18*n + 8)*x^2)*v[i - 2] - ((45*n + 30)*x + (27*n^2 - 36*n - 36)*x^2 - (27*n^2 + 9*n - 6)*x^3)*deriv(v[i - 2], x) + (39*n + 2 + (18*n^2 - 12*n + 2)*x)*v[i - 1] + ((9*n - 3)*x - (9*n - 3)*x^2)*deriv(v[i - 1], x))/(3*n - 1)); 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

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.