Real-rooted atlas

A086645 → D → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(10 - 9\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(20 + 17\,x - 28\,n - 23\,n x + 8\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x + 4\,x^{2} - 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
38210
44836387
5256320964676
612802240120020054925
7614413824107523360360634446
82867278848806403763278405887234227

All 15 rows

Technical details
Node id
n-5204fb736f2a46e2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bfe4d29ca3d34feb

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 -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      0;
   2:      1;
   3:      8,       2;
   4:     48,      36,       3;
   5:    256,     320,      96,       4;
   6:   1280,    2240,    1200,     200,       5;
   7:   6144,   13824,   10752,    3360,     360,      6;
   8:  28672,   78848,   80640,   37632,    7840,    588,     7;
   9: 131072,  425984,  540672,  337920,  107520,  16128,   896,    8;
  10: 589824, 2211840, 3354624, 2635776, 1140480, 266112, 30240, 1296, 9;
%F Formula
The row polynomials satisfy (2*n^2-9*n+10)*P(n,x) = (8*n^2-28*n+20+(6*n^2-23*n+17)*x)*P(n-1,x) - ((4*n-4)*x+(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 28*n + 20 + (6*n^2 - 23*n + 17)*x)*P[n - 1, x] - ((4*n - 4)*x + (4*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 10)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 28*n + 20 + (6*n^2 - 23*n + 17)*x)*v[i - 1] - ((4*n - 4)*x + (4*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 10)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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