Real-rooted atlas

A086645 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(18\,x - 4\,x^{2} + 10\,n - 17\,n x + 2\,n x^{2} - 9\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(40 + 86\,x + 26\,x^{2} - 12\,x^{3} + 4\,n - 53\,n x - 37\,n x^{2} + 6\,n x^{3} - 28\,n^{2} - 7\,n^{2} x + 12\,n^{2} x^{2} + 8\,n^{3} + 6\,n^{3} x\right)\,P_{n-1}(x) + \left(8\,x + 18\,x^{2} + 6\,x^{3} - 4\,x^{4} + 4\,n x - 4\,n x^{2} - 8\,n x^{3} - 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
31610127
48084211186
53845762563611253
61792352023206005518288
7819219968176646944120078154047
836864107520120064645121724821561051348470

All 15 rows

Technical details
Node id
n-4ae2f4817bbdf1ce · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-fbd5bbfbedc0feb7

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      3,       1;
   3:     16,      10,       1;
   4:     80,      84,      21,       1;
   5:    384,     576,     256,      36,       1;
   6:   1792,    3520,    2320,     600,      55,      1;
   7:   8192,   19968,   17664,    6944,    1200,     78,     1;
   8:  36864,  107520,  120064,   64512,   17248,   2156,   105,    1;
   9: 163840,  557056,  753664,  518144,  192000,  37632,  3584,  136,   1;
  10: 720896, 2801664, 4460544, 3753984, 1799424, 494208, 74592, 5616, 171, 1;
%F Formula
The row polynomials satisfy (2*n^3-9*n^2+10*n+(4*n^2-17*n+18)*x+(2*n-4)*x^2)*P(n,x) = (8*n^3-28*n^2+4*n+40+(6*n^3-7*n^2-53*n+86)*x+(12*n^2-37*n+26)*x^2+(6*n-12)*x^3)*P(n-1,x) - ((4*n^2-4*n-8)*x+(4*n^2+4*n-18)*x^2+(8*n-6)*x^3+4*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 28*n^2 + 4*n + 40 + (6*n^3 - 7*n^2 - 53*n + 86)*x + (12*n^2 - 37*n + 26)*x^2 + (6*n - 12)*x^3)*P[n - 1, x] - ((4*n^2 - 4*n - 8)*x + (4*n^2 + 4*n - 18)*x^2 + (8*n - 6)*x^3 + 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 9*n^2 + 10*n + (4*n^2 - 17*n + 18)*x + (2*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = ((8*n^3 - 28*n^2 + 4*n + 40 + (6*n^3 - 7*n^2 - 53*n + 86)*x + (12*n^2 - 37*n + 26)*x^2 + (6*n - 12)*x^3)*v[i - 1] - ((4*n^2 - 4*n - 8)*x + (4*n^2 + 4*n - 18)*x^2 + (8*n - 6)*x^3 + 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 9*n^2 + 10*n + (4*n^2 - 17*n + 18)*x + (2*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, v[i])))};
T(10)

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