Real-rooted atlas

A104684 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-3 - 6\,x - 3\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n}(x) = \left(-41 - 212\,x - 381\,x^{2} - 290\,x^{3} - 80\,x^{4} + 25\,n + 124\,n x + 213\,n x^{2} + 154\,n x^{3} + 40\,n x^{4}\right)\,P_{n-1}(x) + \left(-4 - 38\,x - 130\,x^{2} - 202\,x^{3} - 146\,x^{4} - 40\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2369
3135046109
4633606663961485
5321243667868196360621345
6168315930593101083609693033876316089
789891019584752601163840157695011192283245564770781
848639643020360082811061960201148502162764812718020315189672966861

All 15 rows

Technical details
Node id
n-fcede3bb0c43f0a9 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-04f9880d32434ab5

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,        6;
   3:      13,       50,        46;
   4:      63,      360,       666,       396;
   5:     321,     2436,      6786,      8196,       3606;
   6:    1683,    15930,     59310,    108360,      96930,      33876;
   7:    8989,   101958,    475260,   1163840,    1576950,    1119228,     324556;
   8:   48639,   643020,   3600828,  11061960,   20114850,   21627648,   12718020,    3151896;
   9:  265729,  4012040,  26233228,  96960248,  221412730,  319635848,  284654020,  142866056,   30915046;
  10: 1462563, 24829038, 185659668, 802167408, 2205800478, 4001024916, 4784399676, 3634819056, 1591041906, 305543556;
%F Formula
The row polynomials satisfy (3*n-3+(6*n-6)*x+(3*n-3)*x^2)*P(n,x) = (25*n-41+(124*n-212)*x+(213*n-381)*x^2+(154*n-290)*x^3+(40*n-80)*x^4)*P(n-1,x) - (4+38*x+130*x^2+202*x^3+146*x^4+40*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((25*n - 41 + (124*n - 212)*x + (213*n - 381)*x^2 + (154*n - 290)*x^3 + (40*n - 80)*x^4)*P[n - 1, x] - (4 + 38*x + 130*x^2 + 202*x^3 + 146*x^4 + 40*x^5)*D[P[n - 1, x], x])/(3*n - 3 + (6*n - 6)*x + (3*n - 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((25*n - 41 + (124*n - 212)*x + (213*n - 381)*x^2 + (154*n - 290)*x^3 + (40*n - 80)*x^4)*v[i - 1] - (4 + 38*x + 130*x^2 + 202*x^3 + 146*x^4 + 40*x^5)*deriv(v[i - 1], x))/(3*n - 3 + (6*n - 6)*x + (3*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, v[i])))};
T(10)

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