Real-rooted atlas

A116414 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(8\,n\right)\,P_{n}(x) = \left(-20\,x - 13\,x^{2} + 32\,n + 60\,n x + 13\,n x^{2}\right)\,P_{n-1}(x) + \left(-16\,x - 40\,x^{2} - 13\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1459
216402278
36424027695675
42561280225616404095841
51024640015232171209008176050544
640963072092160140800114864471607573437373
71638414336051916810048001117760711600239068325853784725

All 15 rows

Technical details
Node id
n-ee06806b5d758296 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e93a5d0c12234b5b

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:      4,       5;
  2:     16,      40,       22;
  3:     64,     240,      276,       95;
  4:    256,    1280,     2256,     1640,      409;
  5:   1024,    6400,    15232,    17120,     9008,     1760;
  6:   4096,   30720,    92160,   140800,   114864,    47160,     7573;
  7:  16384,  143360,   519168,  1004800,  1117760,   711600,   239068,    32585;
  8:  65536,  655360,  2781184,  6522880,  9218560,  8012800,  4169680,  1184200,  140206;
  9: 262144, 2949120, 14352384, 39567360, 67931136, 75129600, 53389440, 23450400, 5764620, 603275;
%F Formula
The row polynomials satisfy 8*n*P(n,x) = (32*n+(60*n-20)*x+(13*n-13)*x^2)*P(n-1,x) - (16*x+40*x^2+13*x^3)*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[((32*n + (60*n - 20)*x + (13*n - 13)*x^2)*P[n - 1, x] - (16*x + 40*x^2 + 13*x^3)*D[P[n - 1, x], x])/(8*n)]];
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] = ((32*n + (60*n - 20)*x + (13*n - 13)*x^2)*v[i - 1] - (16*x + 40*x^2 + 13*x^3)*deriv(v[i - 1], x))/(8*n)); 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. [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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023