Real-rooted atlas

A166343 → D

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-2 + 2\,x - 2\,n + n x^{2} - 2\,n^{2} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-x + x^{2} - n x + n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 3\,n + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 12 + 2\,x\)
\(P_{2}(x) = 27 + 54\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
112214
22754384
3583241744560
41211436215448454200
5248551817232110361240635280
6503195301103131470844882530187329280
710146563261702214708401028370196896709883386880
8203721312031572241229037615362970631444874592016296938102400

All 15 rows

Technical details
Node id
n-674032b73934f4eb · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4c4143f463ed4f9a

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 0)
  • 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:   12,       2;
   2:   27,      54,        3;
   3:   58,     324,      174,         4;
   4:  121,    1436,     2154,       484,          5;
   5:  248,    5518,    17232,     11036,       1240,          6;
   6:  503,   19530,   110313,    147084,      48825,       3018,          7;
   7: 1014,   65632,   617022,   1470840,    1028370,     196896,       7098,         8;
   8: 2037,  213120,  3157224,  12290376,   15362970,    6314448,     745920,     16296,       9;
   9: 4084,  676026, 15186336,  90919304,  184355640,  136378956,   35434784,   2704104,   36756,    10;
  10: 8179, 2109758, 69871695, 616584104, 1901677750, 2282013300, 1079022182, 186324520, 9493911, 81790, 11;
%F Formula
The row polynomials satisfy n*P(n,x) = -(2*n+2+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) - ((n+1)*x+(n^2-1)*x^2-(n^2+n)*x^3)*d/dx P(n-2,x) + (3*n+1+2*n^2*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 12 + 2*x, P(2,x) = 27 + 54*x + 3*x^2.
%t Mathematica
P[1, x_] := 12 + 2*x;
P[2, x_] := 27 + 54*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (3*n + 1 + 2*n^2*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 12 + 2*x; v[2] = 27 + 54*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*v[i - 2] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 2], x) + (3*n + 1 + 2*n^2*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], x))/(n)); 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.