Real-rooted atlas

A026729 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2\,x - x^{2} - 4\,n x + 3\,n x^{2} - n^{2} + 9\,n^{2} x - 3\,n^{2} x^{2} - n^{3} + n^{3} x^{2} + n^{4} - 3\,n^{4} x + n^{5}\right)\,P_{n}(x) = \left(4\,x + 2\,x^{2} + 10\,n x - 9\,n x^{2} + 4\,n^{2} - 2\,n^{2} x - 18\,n^{2} x^{2} + 2\,n^{2} x^{3} + 6\,n^{3} + 14\,n^{3} x - 11\,n^{3} x^{2} + 10\,n^{4} x - 2\,n^{5}\right)\,P_{n-1}(x) + \left(-4\,n x^{2} - n x^{3} + 4\,n^{2} x - 23\,n^{2} x^{2} - 2\,n^{2} x^{3} + n^{2} x^{4} + 8\,n^{3} x - 10\,n^{3} x^{2} - 3\,n^{3} x^{3} + 4\,n^{4} x + n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2088
3082735
4005464118
50027192125344
6000192500216908
70006475010803432237
80000500216020585125230

All 15 rows

Technical details
Node id
n-be0a436924532fa1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-0cc1bffa76d861e0

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, 8;
   3: 0, 8, 27;
   4: 0, 0, 54,  64;
   5: 0, 0, 27, 192, 125;
   6: 0, 0,  0, 192, 500,  216;
   7: 0, 0,  0,  64, 750, 1080,  343;
   8: 0, 0,  0,   0, 500, 2160, 2058,   512;
   9: 0, 0,  0,   0, 125, 2160, 5145,  3584,  729;
  10: 0, 0,  0,   0,   0, 1080, 6860, 10752, 5832, 1000;
%F Formula
The row polynomials satisfy (n^5+n^4-n^3-n^2-(3*n^4-9*n^2+4*n+2)*x+(n^3-3*n^2+3*n-1)*x^2)*P(n,x) = -(2*n^5-6*n^3-4*n^2-(10*n^4+14*n^3-2*n^2+10*n+4)*x+(11*n^3+18*n^2+9*n-2)*x^2-2*n^2*x^3)*P(n-1,x) + ((4*n^4+8*n^3+4*n^2)*x+(n^4-10*n^3-23*n^2-4*n)*x^2-(3*n^3+2*n^2+n)*x^3+n^2*x^4)*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^5 - 6*n^3 - 4*n^2 - (10*n^4 + 14*n^3 - 2*n^2 + 10*n + 4)*x + (11*n^3 + 18*n^2 + 9*n - 2)*x^2 - 2*n^2*x^3)*P[n - 1, x] + ((4*n^4 + 8*n^3 + 4*n^2)*x + (n^4 - 10*n^3 - 23*n^2 - 4*n)*x^2 - (3*n^3 + 2*n^2 + n)*x^3 + n^2*x^4)*D[P[n - 1, x], x])/(n^5 + n^4 - n^3 - n^2 - (3*n^4 - 9*n^2 + 4*n + 2)*x + (n^3 - 3*n^2 + 3*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^5 - 6*n^3 - 4*n^2 - (10*n^4 + 14*n^3 - 2*n^2 + 10*n + 4)*x + (11*n^3 + 18*n^2 + 9*n - 2)*x^2 - 2*n^2*x^3)*v[i - 1] + ((4*n^4 + 8*n^3 + 4*n^2)*x + (n^4 - 10*n^3 - 23*n^2 - 4*n)*x^2 - (3*n^3 + 2*n^2 + n)*x^3 + n^2*x^4)*deriv(v[i - 1], x))/(n^5 + n^4 - n^3 - n^2 - (3*n^4 - 9*n^2 + 4*n + 2)*x + (n^3 - 3*n^2 + 3*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

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.