Real-rooted atlas

A028338 → basis A008277 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2814325
348114454211
4384112262110452236
538401311091352180200628471
6460801781101475554488059253427422899
7645120276381026280459597001627501360853987173580

All 15 rows

Technical details
Node id
n-7374db76fd3c4567 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ec3729951d34686d

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         2,         2;
  2:         8,        14,          3;
  3:        48,       114,         45,         4;
  4:       384,      1122,        621,       104,         5;
  5:      3840,     13110,       9135,      2180,       200,        6;
  6:     46080,    178110,     147555,     44880,      5925,      342,       7;
  7:    645120,   2763810,    2628045,    959700,    162750,    13608,     539,     8;
  8:  10321920,  48275010,   51450525,  21738360,   4454625,   481068,   27734,   800,    9;
  9: 185794560, 937869030, 1101423015, 525106260, 125174700, 16410114, 1226862, 51696, 1134, 10;
%F Formula
The row polynomials satisfy (2-2*x+2*n*x^2)*P(n,x) = (4*n-(4*n-4)*x+(4*n^2-4)*x^2+(2*n+2)*x^3)*P(n-1,x) + (2*x-3*x^2+(2*n+3)*x^3)*d/dx P(n-1,x) + x^3*d^2/dx^2 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[Together[((4*n - (4*n - 4)*x + (4*n^2 - 4)*x^2 + (2*n + 2)*x^3)*P[n - 1, x] + (2*x - 3*x^2 + (2*n + 3)*x^3)*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}])/(2 - 2*x + 2*n*x^2)]];
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] = ((4*n - (4*n - 4)*x + (4*n^2 - 4)*x^2 + (2*n + 2)*x^3)*v[i - 1] + (2*x - 3*x^2 + (2*n + 3)*x^3)*deriv(v[i - 1], x) + x^3*deriv(deriv(v[i - 1], x), x))/(2 - 2*x + 2*n*x^2)); 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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.