Real-rooted atlas

A086645 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(10 - 9\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(20 + 51\,x - 28\,n - 69\,n x + 8\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} - 4\,n x - 12\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
233
3241842
414432481549
5768288025923246564
63840201603240016200121573815
718432124416290304272160874804374797166
886016709632217728030481921905120428652153098370201

All 15 rows

Technical details
Node id
n-b6cf9b6d8bf8869b · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-05f5318533cf2c44

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:       3;
   3:      24,       18;
   4:     144,      324,       81;
   5:     768,     2880,     2592,       324;
   6:    3840,    20160,    32400,     16200,      1215;
   7:   18432,   124416,   290304,    272160,     87480,      4374;
   8:   86016,   709632,  2177280,   3048192,   1905120,    428652,    15309;
   9:  393216,  3833856, 14598144,  27371520,  26127360,  11757312,  1959552,   52488;
  10: 1769472, 19906560, 90574848, 213497856, 277136640, 193995648, 66134880, 8503056, 177147;
%F Formula
The row polynomials satisfy (2*n^2-9*n+10)*P(n,x) = (8*n^2-28*n+20+(18*n^2-69*n+51)*x)*P(n-1,x) - ((4*n-4)*x+(12*n-12)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 28*n + 20 + (18*n^2 - 69*n + 51)*x)*P[n - 1, x] - ((4*n - 4)*x + (12*n - 12)*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] = 3; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 28*n + 20 + (18*n^2 - 69*n + 51)*x)*v[i - 1] - ((4*n - 4)*x + (12*n - 12)*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