Real-rooted atlas

A103327 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 13\,x - 4\,x^{2} + 16\,n - 34\,n x + 8\,n x^{2} - 20\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(40\,x + 164\,x^{2} - 80\,x^{3} + 16\,n - 32\,n x - 344\,n x^{2} + 96\,n x^{3} - 48\,n^{2} - 144\,n^{2} x + 192\,n^{2} x^{2} + 32\,n^{3} + 96\,n^{3} x\right)\,P_{n-1}(x) + \left(20\,x^{2} + 64\,x^{3} - 64\,x^{4} + 8\,n x - 128\,n x^{3} - 16\,n^{2} x - 64\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
2648016160
33841088576642112
42048947211520358425626880
5102406656013619297280204801024331776
6491524136961216512149094471680011059240964001792
722937623756809191424164659201376256048168965734401638447431680

All 15 rows

Technical details
Node id
n-f783cab6fae5f997 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-285a8f351a67f82b

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:       8,        4;
  2:      64,       80,        16;
  3:     384,     1088,       576,         64;
  4:    2048,     9472,     11520,       3584,        256;
  5:   10240,    66560,    136192,      97280,      20480,       1024;
  6:   49152,   413696,   1216512,    1490944,     716800,     110592,      4096;
  7:  229376,  2375680,   9191424,   16465920,   13762560,    4816896,    573440,     16384;
  8: 1048576, 12910592,  62193664,  148504576,  183828480,  112852992,  30277632,   2883584,    65536;
  9: 4718592, 67371008, 389283840, 1168900096, 1949302784, 1790705664, 847773696, 180879360, 14155776, 262144;
%F Formula
The row polynomials satisfy (8*n^3-20*n^2+16*n-4+(16*n^2-34*n+13)*x+(8*n-4)*x^2)*P(n,x) = (32*n^3-48*n^2+16*n+(96*n^3-144*n^2-32*n+40)*x+(192*n^2-344*n+164)*x^2+(96*n-80)*x^3)*P(n-1,x) - ((16*n^2-8*n)*x+(64*n^2-20)*x^2+(128*n-64)*x^3+64*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[((32*n^3 - 48*n^2 + 16*n + (96*n^3 - 144*n^2 - 32*n + 40)*x + (192*n^2 - 344*n + 164)*x^2 + (96*n - 80)*x^3)*P[n - 1, x] - ((16*n^2 - 8*n)*x + (64*n^2 - 20)*x^2 + (128*n - 64)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^2 - 34*n + 13)*x + (8*n - 4)*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] = ((32*n^3 - 48*n^2 + 16*n + (96*n^3 - 144*n^2 - 32*n + 40)*x + (192*n^2 - 344*n + 164)*x^2 + (96*n - 80)*x^3)*v[i - 1] - ((16*n^2 - 8*n)*x + (64*n^2 - 20)*x^2 + (128*n - 64)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^2 - 34*n + 13)*x + (8*n - 4)*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

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