Real-rooted atlas

A103327 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 19\,x - 6\,x^{2} + 21\,n - 50\,n x + 12\,n x^{2} - 28\,n^{2} + 24\,n^{2} x + 12\,n^{3}\right)\,P_{n}(x) = \left(4 + 70\,x + 171\,x^{2} - 90\,x^{3} + 12\,n - 107\,n x - 366\,n x^{2} + 108\,n x^{3} - 64\,n^{2} - 120\,n^{2} x + 216\,n^{2} x^{2} + 48\,n^{3} + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x + 34\,x^{2} + 60\,x^{3} - 72\,x^{4} + 8\,n x - 24\,n x^{2} - 144\,n x^{3} - 24\,n^{2} x - 72\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17310
252549115
3304672297271300
416005664540014048114149
579363916861488345606075243149470
637888241152539136508032192780247867291544503
7176128137625640296965495040353808097977696957218715694120

All 15 rows

Technical details
Node id
n-57c180ed1abc870d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bb664eaedc75bc11

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:       7,        3;
  2:      52,       54,         9;
  3:     304,      672,       297,        27;
  4:    1600,     5664,      5400,      1404,        81;
  5:    7936,    39168,     61488,     34560,      6075,       243;
  6:   37888,   241152,    539136,    508032,    192780,     24786,       729;
  7:  176128,  1376256,   4029696,   5495040,   3538080,    979776,     96957,     2187;
  8:  802816,  7446528,  27076608,  48964608,  46189440,  21881664,   4653936,   367416,    6561;
  9: 3604480, 38731776, 168652800, 382427136, 483335424, 338722560, 123941664, 20995200, 1358127, 19683;
%F Formula
The row polynomials satisfy (12*n^3-28*n^2+21*n-5+(24*n^2-50*n+19)*x+(12*n-6)*x^2)*P(n,x) = (48*n^3-64*n^2+12*n+4+(108*n^3-120*n^2-107*n+70)*x+(216*n^2-366*n+171)*x^2+(108*n-90)*x^3)*P(n-1,x) - ((24*n^2-8*n-2)*x+(72*n^2+24*n-34)*x^2+(144*n-60)*x^3+72*x^4)*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[((48*n^3 - 64*n^2 + 12*n + 4 + (108*n^3 - 120*n^2 - 107*n + 70)*x + (216*n^2 - 366*n + 171)*x^2 + (108*n - 90)*x^3)*P[n - 1, x] - ((24*n^2 - 8*n - 2)*x + (72*n^2 + 24*n - 34)*x^2 + (144*n - 60)*x^3 + 72*x^4)*D[P[n - 1, x], x])/(12*n^3 - 28*n^2 + 21*n - 5 + (24*n^2 - 50*n + 19)*x + (12*n - 6)*x^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] = ((48*n^3 - 64*n^2 + 12*n + 4 + (108*n^3 - 120*n^2 - 107*n + 70)*x + (216*n^2 - 366*n + 171)*x^2 + (108*n - 90)*x^3)*v[i - 1] - ((24*n^2 - 8*n - 2)*x + (72*n^2 + 24*n - 34)*x^2 + (144*n - 60)*x^3 + 72*x^4)*deriv(v[i - 1], x))/(12*n^3 - 28*n^2 + 21*n - 5 + (24*n^2 - 50*n + 19)*x + (12*n - 6)*x^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

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