Real-rooted atlas

A113214 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(4 + 18\,x + 29\,x^{2} + 20\,x^{3} + 5\,x^{4} - 2\,n - 10\,n x - 19\,n x^{2} - 16\,n x^{3} - 5\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 17\,x^{2} + 27\,x^{3} + 19\,x^{4} + 5\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
206915
304202044
400155040105
50065412078258
600028154266147595
700081124085682721368

All 15 rows

Technical details
Node id
n-a1cb8cca853be542 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c12a5875baa63cc4

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: 2, 4;
  2: 0, 6,  9;
  3: 0, 4, 20, 20;
  4: 0, 0, 15, 50,  40;
  5: 0, 0,  6, 54, 120,  78;
  6: 0, 0,  0, 28, 154, 266,  147;
  7: 0, 0,  0,  8, 112, 408,  568,  272;
  8: 0, 0,  0,  0,  45, 360,  999, 1170,  495;
  9: 0, 0,  0,  0,  10, 200, 1050, 2330, 2350, 890;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(2*n-4+(10*n-18)*x+(19*n-29)*x^2+(16*n-20)*x^3+(5*n-5)*x^4)*P(n-1,x) + (4*x+17*x^2+27*x^3+19*x^4+5*x^5)*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 - 4 + (10*n - 18)*x + (19*n - 29)*x^2 + (16*n - 20)*x^3 + (5*n - 5)*x^4)*P[n - 1, x] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + 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] = (-(2*n - 4 + (10*n - 18)*x + (19*n - 29)*x^2 + (16*n - 20)*x^3 + (5*n - 5)*x^4)*v[i - 1] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + 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

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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.