Real-rooted atlas

A039755 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(-4 - 6\,x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(6 + 2\,x\right)\,P_{n-1}(x) + \left(2 + 2\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3628
43424361
5208224604496
61396200084012054361
71023218152104402360210641400
8813201711361249083920055303367422437

All 15 rows

Technical details
Node id
n-ee3ca24b971c94de · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e6aa5e2f05ec8603

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:       1;
   3:       6,        2;
   4:      34,       24,        3;
   5:     208,      224,       60,       4;
   6:    1396,     2000,      840,     120,       5;
   7:   10232,    18152,    10440,    2360,     210,      6;
   8:   81320,   171136,   124908,   39200,    5530,    336,     7;
   9:  696192,  1689088,  1491168,  602784,  118720,  11424,   504,   8;
  10: 6382864, 17491584, 18079680, 8965824, 2292360, 308448, 21504, 720, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(6+4*x+x^2)*P(n-2,x) - (4+6*x+2*x^2)*d/dx P(n-2,x) + (6+2*x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(6 + 4*x + x^2)*P[n - 2, x] - (4 + 6*x + 2*x^2)*D[P[n - 2, x], x] + (6 + 2*x)*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(6 + 4*x + x^2)*v[i - 2] - (4 + 6*x + 2*x^2)*deriv(v[i - 2], x) + (6 + 2*x)*v[i - 1] + (2 + 2*x)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.