Real-rooted atlas

A114687 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(8 - 8\,x + 2\,x^{2} + 2\,n + n x^{2} + n^{2} + 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(16 - 16\,x + 4\,x^{2} + 12\,n + 4\,n x - 7\,n x^{2} + 3\,n x^{3} + 6\,n^{2} + 3\,n^{2} x + 6\,n^{2} x^{2} + 2\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(8\,x - 12\,x^{2} + 8\,x^{3} - 2\,x^{4} + 2\,n x + 7\,n x^{2} - 4\,n x^{3} + 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2108119
3324815196
496240140241501
5272104010003203512668
673640325880308063048114407
719201433629792235207840112063178592

All 15 rows

Technical details
Node id
n-e09489fb7491a8fa · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-48b123f09be537c0

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:     3,      1;
  2:    10,      8,      1;
  3:    32,     48,     15,      1;
  4:    96,    240,    140,     24,      1;
  5:   272,   1040,   1000,    320,     35,      1;
  6:   736,   4032,   5880,   3080,    630,     48,     1;
  7:  1920,  14336,  29792,  23520,   7840,   1120,    63,    1;
  8:  4864,  47616, 134400, 150528,  75264,  17472,  1848,   80,  1;
  9: 12032, 149760, 552960, 838656, 592704, 205632, 35280, 2880, 99, 1;
%F Formula
The row polynomials satisfy (n^3+n^2+2*n+8+(2*n^2-8)*x+(n+2)*x^2)*P(n,x) = (2*n^3+6*n^2+12*n+16+(3*n^3+3*n^2+4*n-16)*x+(6*n^2-7*n+4)*x^2+3*n*x^3)*P(n-1,x) + ((4*n^2+2*n+8)*x-(2*n^2-7*n+12)*x^2-(4*n-8)*x^3-2*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[((2*n^3 + 6*n^2 + 12*n + 16 + (3*n^3 + 3*n^2 + 4*n - 16)*x + (6*n^2 - 7*n + 4)*x^2 + 3*n*x^3)*P[n - 1, x] + ((4*n^2 + 2*n + 8)*x - (2*n^2 - 7*n + 12)*x^2 - (4*n - 8)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 + n^2 + 2*n + 8 + (2*n^2 - 8)*x + (n + 2)*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] = ((2*n^3 + 6*n^2 + 12*n + 16 + (3*n^3 + 3*n^2 + 4*n - 16)*x + (6*n^2 - 7*n + 4)*x^2 + 3*n*x^3)*v[i - 1] + ((4*n^2 + 2*n + 8)*x - (2*n^2 - 7*n + 12)*x^2 - (4*n - 8)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 + n^2 + 2*n + 8 + (2*n^2 - 8)*x + (n + 2)*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. [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