Real-rooted atlas

A091044 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(1 - 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2\,x - 4\,n - 14\,n x + 8\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2322456
319232096608
41024268819203205952
551201843221504896096054976
624576112640184320129024358402688489088
71146886389761351680135168064512012902471684238336

All 15 rows

Technical details
Node id
n-96a96938cda4aada · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c015fe9c958530bb

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:       4;
  2:      32,       24;
  3:     192,      320,       96;
  4:    1024,     2688,     1920,      320;
  5:    5120,    18432,    21504,     8960,      960;
  6:   24576,   112640,   184320,   129024,    35840,     2688;
  7:  114688,   638976,  1351680,  1351680,   645120,   129024,     7168;
  8:  524288,  3440640,  8945664, 11714560,  8110080,  2838528,   430080,   18432;
  9: 2359296, 17825792, 55050240, 89456640, 82001920, 42172416, 11354112, 1351680, 46080;
%F Formula
The row polynomials satisfy (2*n^2-3*n+1)*P(n,x) = (8*n^2-4*n+(12*n^2-14*n-2)*x)*P(n-1,x) - (4*n*x+8*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 4*n + (12*n^2 - 14*n - 2)*x)*P[n - 1, x] - (4*n*x + 8*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 4*n + (12*n^2 - 14*n - 2)*x)*v[i - 1] - (4*n*x + 8*n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 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

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