Real-rooted atlas

A327997 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-54 + 45\,n - 9\,n^{2}\right)\,P_{n-4}(x) + \left(-6 - 12\,x - 9\,n + 6\,n x + 6\,n^{2}\right)\,P_{n-3}(x) + \left(8 - 2\,x - x^{2} - 8\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(3 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 7 + 2\,x\)
\(P_{3}(x) = 38 + 24\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2729
33824365
4192202544452
5969151064510053229
651151098866151600165624489
72932281942644072142033952527200745

All 15 rows

Technical details
Node id
n-73a2b4ae4a756a65 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6d4c954106f7ca82

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 -1)
  • 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:       0;
  1:       1;
  2:       7,       2;
  3:      38,      24,       3;
  4:     192,     202,      54,       4;
  5:     969,    1510,     645,     100,      5;
  6:    5115,   10988,    6615,    1600,    165,      6;
  7:   29322,   81942,   64407,   21420,   3395,    252,     7;
  8:  187992,  647316,  628068,  266476,  57260,   6468,   364,   8;
  9: 1370745, 5529852, 6333954, 3254580, 881475, 133812, 11382, 504, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-9*n-6+(6*n-12)*x)*P(n-3,x) - (n^2+8*n-8+(2*n+2)*x+x^2)*P(n-2,x) + (2*n+3+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 7 + 2*x, P(3,x) = 38 + 24*x + 3*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 7 + 2*x;
P[3, x_] := 38 + 24*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 9*n - 6 + (6*n - 12)*x)*P[n - 3, x] - (n^2 + 8*n - 8 + (2*n + 2)*x + x^2)*P[n - 2, x] + (2*n + 3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 7 + 2*x; v[4] = 38 + 24*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(9*n^2 - 45*n + 54)*v[i - 4] + (6*n^2 - 9*n - 6 + (6*n - 12)*x)*v[i - 3] - (n^2 + 8*n - 8 + (2*n + 2)*x + x^2)*v[i - 2] + (2*n + 3 + 2*x)*v[i - 1]); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023