Real-rooted atlas

A133087 → reverse → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-2 - 6\,x - 3\,x^{2} + 5\,n + 8\,n x + 3\,n x^{2}\right)\,P_{n-1}(x) + \left(-10 - 21\,x - 14\,x^{2} - 3\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
24419
3122011245
41632248181
54812813672192405
66419224016060121729
719270411049605001562723645

All 15 rows

Technical details
Node id
n-ed8d3658ba94cefd · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-0fc72d33f55fbdc4

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   3,    2;
  2:   4,    4,    1;
  3:  12,   20,   11,    2;
  4:  16,   32,   24,    8,    1;
  5:  48,  128,  136,   72,   19,    2;
  6:  64,  192,  240,  160,   60,   12,    1;
  7: 192,  704, 1104,  960,  500,  156,   27,   2;
  8: 256, 1024, 1792, 1792, 1120,  448,  112,  16,  1;
  9: 768, 3584, 7424, 8960, 6944, 3584, 1232, 272, 35, 2;
%F Formula
The row polynomials satisfy P(n,x) = (5*n-2+(8*n-6)*x+(3*n-3)*x^2)*P(n-1,x) - (10+21*x+14*x^2+3*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5*n - 2 + (8*n - 6)*x + (3*n - 3)*x^2)*P[n - 1, x] - (10 + 21*x + 14*x^2 + 3*x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5*n - 2 + (8*n - 6)*x + (3*n - 3)*x^2)*v[i - 1] - (10 + 21*x + 14*x^2 + 3*x^3)*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.