Real-rooted atlas

A062145 → D → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
244
3302050
416821060438
584015128401403332
6396092407560252028023560
7180185148055440277206300504159462
88008027027036036024024083160138608401048810

All 15 rows

Technical details
Node id
n-b2e9abf9eed0ddf3 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ecca132484b838e2

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:       4;
   3:      30,      20;
   4:     168,     210,       60;
   5:     840,    1512,      840,      140;
   6:    3960,    9240,     7560,     2520,     280;
   7:   18018,   51480,    55440,    27720,    6300,     504;
   8:   80080,  270270,   360360,   240240,   83160,   13860,    840;
   9:  350064, 1361360,  2162160,  1801800,  840840,  216216,  27720,  1320;
  10: 1511640, 6651216, 12252240, 12252240, 7207200, 2522520, 504504, 51480, 1980;
%F Formula
The row polynomials satisfy (n^2-n-2)*P(n,x) = (4*n^2-2*n+(3*n^2-3*n+2)*x)*P(n-1,x) - (2*n*x+2*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n + (3*n^2 - 3*n + 2)*x)*P[n - 1, x] - (2*n*x + 2*n*x^2)*D[P[n - 1, x], x])/(n^2 - n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 2*n + (3*n^2 - 3*n + 2)*x)*v[i - 1] - (2*n*x + 2*n*x^2)*deriv(v[i - 1], x))/(n^2 - n - 2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.