Real-rooted atlas

A207824 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3\,n - 2\,n x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(7 - 2\,x - 10\,x^{2} - 4\,x^{3} + 19\,n + 6\,n x - 10\,n x^{2} - 4\,n x^{3} + 18\,n^{2} + 12\,n^{2} x + 6\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-4 - 22\,x - 34\,x^{2} - 20\,x^{3} - 4\,x^{4} + 8\,n + 12\,n x + 4\,n x^{2} + 8\,n^{2} + 12\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110414
2838016179
36281000480642172
44501101288016256025625461
5311349102010176053504128001024291242
621002375734411002568166403212806144040963271079
7139056859716641070457610322560572928018001922867201638436221944

All 15 rows

Technical details
Node id
n-5687ee62d10ea2e3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-52d6d48c0a0a1a2c

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:       10,         4;
  2:       83,        80,        16;
  3:      628,      1000,       480,         64;
  4:     4501,     10128,      8016,       2560,        256;
  5:    31134,     91020,    101760,      53504,      12800,      1024;
  6:   210023,    757344,   1100256,     816640,     321280,     61440,      4096;
  7:  1390568,   5971664,  10704576,   10322560,    5729280,   1800192,    286720,    16384;
  8:  9075049,  45255520,  96646880,  114998784,   82901760,  36728832,   9605120,  1310720,   65536;
  9: 58548146, 332673428, 825303936, 1170085120, 1040718336, 598692864, 220659712, 49414144, 5898240, 262144;
%F Formula
The row polynomials satisfy (2*n^3+3*n-2*n*x^2)*P(n,x) = (6*n^3+18*n^2+19*n+7+(4*n^3+12*n^2+6*n-2)*x-(10*n+10)*x^2-(4*n+4)*x^3)*P(n-1,x) + (8*n^2+8*n-4+(12*n^2+12*n-22)*x+(4*n^2+4*n-34)*x^2-20*x^3-4*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[((6*n^3 + 18*n^2 + 19*n + 7 + (4*n^3 + 12*n^2 + 6*n - 2)*x - (10*n + 10)*x^2 - (4*n + 4)*x^3)*P[n - 1, x] + (8*n^2 + 8*n - 4 + (12*n^2 + 12*n - 22)*x + (4*n^2 + 4*n - 34)*x^2 - 20*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 + 3*n - 2*n*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] = ((6*n^3 + 18*n^2 + 19*n + 7 + (4*n^3 + 12*n^2 + 6*n - 2)*x - (10*n + 10)*x^2 - (4*n + 4)*x^3)*v[i - 1] + (8*n^2 + 8*n - 4 + (12*n^2 + 12*n - 22)*x + (4*n^2 + 4*n - 34)*x^2 - 20*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 + 3*n - 2*n*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