Real-rooted atlas

A207823 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-28\,n - 10\,n x + 6\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-70 - 88\,x - 30\,x^{2} - 70\,n - 109\,n x - 30\,n x^{2} + 9\,n x^{3} - 21\,n^{2} x + 15\,n^{2} x^{2} + 18\,n^{2} x^{3} + 15\,n^{3} x^{2} + 9\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-49 - 105\,x - 71\,x^{2} - 15\,x^{3} + 21\,n x^{2} + 30\,n x^{3} + 9\,n x^{4} + 21\,n^{2} x^{2} + 30\,n^{2} x^{3} + 9\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15611
2246027111
31154384051081066
45512820396921604059905
5264016968321302656810125145889889
61264997860233037259200149850437405103801439
760605548226157342522015801711125756702178605174967047764

All 15 rows

Technical details
Node id
n-810d48f7bb8a6272 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f8f91a6f973b0e93

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:       5,        6;
  2:      24,       60,       27;
  3:     115,      438,      405,       108;
  4:     551,     2820,     3969,      2160,       405;
  5:    2640,    16968,    32130,     26568,     10125,      1458;
  6:   12649,    97860,   233037,    259200,    149850,     43740,     5103;
  7:   60605,   548226,  1573425,   2201580,   1711125,    756702,   178605,    17496;
  8:  290376,  3006900, 10101105,  17042400,  16661700,   9899820,  3536379,   699840,   59049;
  9: 1391275, 16228530, 62463150, 123414840, 145506375, 108724518, 52152660, 15606432, 2657205, 196830;
%F Formula
The row polynomials satisfy -(28*n+10*n*x-6*n^3*x^2)*P(n,x) = -(70*n+70+(21*n^2+109*n+88)*x-(15*n^3+15*n^2-30*n-30)*x^2-(9*n^3+18*n^2+9*n)*x^3)*P(n-1,x) - (49+105*x-(21*n^2+21*n-71)*x^2-(30*n^2+30*n-15)*x^3-(9*n^2+9*n)*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[(-(70*n + 70 + (21*n^2 + 109*n + 88)*x - (15*n^3 + 15*n^2 - 30*n - 30)*x^2 - (9*n^3 + 18*n^2 + 9*n)*x^3)*P[n - 1, x] - (49 + 105*x - (21*n^2 + 21*n - 71)*x^2 - (30*n^2 + 30*n - 15)*x^3 - (9*n^2 + 9*n)*x^4)*D[P[n - 1, x], x])/(-28*n - 10*n*x + 6*n^3*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] = (-(70*n + 70 + (21*n^2 + 109*n + 88)*x - (15*n^3 + 15*n^2 - 30*n - 30)*x^2 - (9*n^3 + 18*n^2 + 9*n)*x^3)*v[i - 1] - (49 + 105*x - (21*n^2 + 21*n - 71)*x^2 - (30*n^2 + 30*n - 15)*x^3 - (9*n^2 + 9*n)*x^4)*deriv(v[i - 1], x))/(-28*n - 10*n*x + 6*n^3*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023