Real-rooted atlas

A069466 → γ → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(8 + 8\,x - 26\,n - 24\,n x + 20\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(12 + 28\,x + 16\,x^{2} - 24\,n - 56\,n x - 32\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
122
2181230
3140120260
4133016804203430
51285220160756040572
613028424948013860018480536844
71348776302702421621604804807018440

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(n^{2}\right)\,P_{n}(x) = \left(36 + 48\,x - 96\,n - 128\,n x + 48\,n^{2} + 64\,n^{2} x\right)\,P_{n-2}(x) + \left(2 - 8\,n + 8\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
Technical details
Node id
n-c1e8f5a3e0c5db28 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-86c00447f74e63c7)
Fingerprint
f6-24cf4d809f9b6fe7

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:        18,        12;
  3:       140,       120;
  4:      1330,      1680,       420;
  5:     12852,     20160,      7560;
  6:    130284,    249480,    138600,     18480;
  7:   1348776,   3027024,   2162160,    480480;
  8:  14247090,  36756720,  32432400,  10810800,   900900;
  9: 152618180, 444581280, 465585120, 204204000, 30630600;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (20*n^2-26*n+8+(16*n^2-24*n+8)*x)*P(n-1,x) - (24*n-12+(56*n-28)*x+(32*n-16)*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[((20*n^2 - 26*n + 8 + (16*n^2 - 24*n + 8)*x)*P[n - 1, x] - (24*n - 12 + (56*n - 28)*x + (32*n - 16)*x^2)*D[P[n - 1, x], x])/(n^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] = ((20*n^2 - 26*n + 8 + (16*n^2 - 24*n + 8)*x)*v[i - 1] - (24*n - 12 + (56*n - 28)*x + (32*n - 16)*x^2)*deriv(v[i - 1], x))/(n^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

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