Real-rooted atlas

A026729 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(4 + 22\,x + 42\,x^{2} + 34\,x^{3} + 10\,x^{4} - 2\,n - 10\,n x - 19\,n x^{2} - 16\,n x^{3} - 5\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 17\,x^{2} + 27\,x^{3} + 19\,x^{4} + 5\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2022
30257
40061016
5003182041
600012443894
700044210271219
8000020120222130492
900005803154662351101
1000000302607689484202426
11000006135760178218847445311
1200000042490204439723672130811528
1300000072101575518085807044228524881
140000000568404620125201806013332397053398
15000000083082940126702914037206249466865114083
Technical details
Node id
n-d0bf5f74983ec343 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e7a207c59c800cef

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: 0, 2;
   3: 0, 2, 5;
   4: 0, 0, 6, 10;
   5: 0, 0, 3, 18, 20;
   6: 0, 0, 0, 12, 44,  38;
   7: 0, 0, 0,  4, 42, 102,  71;
   8: 0, 0, 0,  0, 20, 120, 222, 130;
   9: 0, 0, 0,  0,  5,  80, 315, 466, 235;
  10: 0, 0, 0,  0,  0,  30, 260, 768, 948, 420;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = -(2*n-4+(10*n-22)*x+(19*n-42)*x^2+(16*n-34)*x^3+(5*n-10)*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 >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (10*n - 22)*x + (19*n - 42)*x^2 + (16*n - 34)*x^3 + (5*n - 10)*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 - 1 + (2*n - 2)*x + (n - 1)*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] = (-(2*n - 4 + (10*n - 22)*x + (19*n - 42)*x^2 + (16*n - 34)*x^3 + (5*n - 10)*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 - 1 + (2*n - 2)*x + (n - 1)*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

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.