Real-rooted atlas

A033282 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\,n + n^{2}\right)\,P_{n}(x) = \left(2 - 2\,x + 6\,n + 2\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1415
21510126
35663181138
4210336168281743
5792165012003604014043
6300377227425330067554122180
71144035035420422502577001155701122468

All 15 rows

Technical details
Node id
n-09c4d3901836573e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6832a290ef8b97c2

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:      4,      1;
  2:     15,     10,       1;
  3:     56,     63,      18,       1;
  4:    210,    336,     168,      28,      1;
  5:    792,   1650,    1200,     360,     40,      1;
  6:   3003,   7722,    7425,    3300,    675,     54,     1;
  7:  11440,  35035,   42042,   25025,   7700,   1155,    70,    1;
  8:  43758, 155584,  224224,  168168,  70070,  16016,  1848,   88,   1;
  9: 167960, 680238, 1145664, 1039584, 550368, 171990, 30576, 2808, 108, 1;
%F Formula
The row polynomials satisfy (n^2+2*n)*P(n,x) = (4*n^2+6*n+2+(3*n^2+2*n-2)*x)*P(n-1,x) - ((2*n+2)*x+(2*n+2)*x^2)*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[Together[((4*n^2 + 6*n + 2 + (3*n^2 + 2*n - 2)*x)*P[n - 1, x] - ((2*n + 2)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n)]];
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^2 + 6*n + 2 + (3*n^2 + 2*n - 2)*x)*v[i - 1] - ((2*n + 2)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023