Real-rooted atlas

A099093 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9\,n - 6\,n x + 9\,n^{2} + 9\,n^{3} x + n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n}(x) = \left(-72\,x - 12\,x^{2} - 8\,x^{3} + 27\,n - 18\,n x - 42\,n x^{2} - 8\,n x^{3} + 27\,n^{2} + 72\,n^{2} x + 24\,n^{2} x^{2} + 6\,n^{2} x^{3} + 27\,n^{3} x + 33\,n^{3} x^{2} + 8\,n^{3} x^{3} + 3\,n^{4} x^{2} + 2\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-27\,x - 54\,x^{2} - 36\,x^{3} - 16\,x^{4} - 27\,n x - 90\,n x^{2} - 96\,n x^{3} - 32\,n x^{4} - 27\,n^{2} x^{2} - 51\,n^{2} x^{3} - 20\,n^{2} x^{4} - 3\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
133
291221
3277299
48132481486
524312967292268
67294860437443210395
721871749621870518446737

All 15 rows

Technical details
Node id
n-4090270705fd9166 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-f79258839f63af14

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:     9,     12;
  3:    27,     72;
  4:    81,    324,     81;
  5:   243,   1296,    729;
  6:   729,   4860,   4374,    432;
  7:  2187,  17496,  21870,   5184;
  8:  6561,  61236,  98415,  38880,  2025;
  9: 19683, 209952, 413343, 233280, 30375;
%F Formula
The row polynomials satisfy (9*n^2+9*n+(9*n^3-6*n)*x+(n^4+n^3)*x^2)*P(n,x) = (27*n^2+27*n+(27*n^3+72*n^2-18*n-72)*x+(3*n^4+33*n^3+24*n^2-42*n-12)*x^2+(2*n^4+8*n^3+6*n^2-8*n-8)*x^3)*P(n-1,x) - ((27*n+27)*x+(27*n^2+90*n+54)*x^2+(3*n^3+51*n^2+96*n+36)*x^3+(4*n^3+20*n^2+32*n+16)*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[((27*n^2 + 27*n + (27*n^3 + 72*n^2 - 18*n - 72)*x + (3*n^4 + 33*n^3 + 24*n^2 - 42*n - 12)*x^2 + (2*n^4 + 8*n^3 + 6*n^2 - 8*n - 8)*x^3)*P[n - 1, x] - ((27*n + 27)*x + (27*n^2 + 90*n + 54)*x^2 + (3*n^3 + 51*n^2 + 96*n + 36)*x^3 + (4*n^3 + 20*n^2 + 32*n + 16)*x^4)*D[P[n - 1, x], x])/(9*n^2 + 9*n + (9*n^3 - 6*n)*x + (n^4 + n^3)*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] = ((27*n^2 + 27*n + (27*n^3 + 72*n^2 - 18*n - 72)*x + (3*n^4 + 33*n^3 + 24*n^2 - 42*n - 12)*x^2 + (2*n^4 + 8*n^3 + 6*n^2 - 8*n - 8)*x^3)*v[i - 1] - ((27*n + 27)*x + (27*n^2 + 90*n + 54)*x^2 + (3*n^3 + 51*n^2 + 96*n + 36)*x^3 + (4*n^3 + 20*n^2 + 32*n + 16)*x^4)*deriv(v[i - 1], x))/(9*n^2 + 9*n + (9*n^3 - 6*n)*x + (n^4 + 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 - 1, v[i])))};
T(9)

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.