Real-rooted atlas

A119900 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 3\,n^{2} + 2\,n^{2} x - n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-2 + 4\,x - 2\,x^{2} - n - 5\,n x - 4\,n x^{2} + 2\,n^{2} - 16\,n^{2} x + 18\,n^{2} x^{2} + n^{3} - 7\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x + 8\,x^{2} - 4\,x^{3} - 6\,n x + 22\,n x^{2} - 16\,n x^{3} - 2\,n^{2} x + 10\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1044
202911
300121628
4003402568
50002410036160
6000410521049368
700004033639264832

All 15 rows

Technical details
Node id
n-791df8959e4bc780 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8dc4210127e6aef6

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: 0, 4;
  2: 0, 2,  9;
  3: 0, 0, 12, 16;
  4: 0, 0,  3, 40,  25;
  5: 0, 0,  0, 24, 100,  36;
  6: 0, 0,  0,  4, 105, 210,  49;
  7: 0, 0,  0,  0,  40, 336, 392,   64;
  8: 0, 0,  0,  0,   5, 216, 882,  672,   81;
  9: 0, 0,  0,  0,   0,  60, 840, 2016, 1080, 100;
%F Formula
The row polynomials satisfy -(n^3+3*n^2+2*n-(4*n^3+2*n^2)*x)*P(n,x) = (n^3+2*n^2-n-2-(7*n^3+16*n^2+5*n-4)*x+(12*n^3+18*n^2-4*n-2)*x^2)*P(n-1,x) - ((2*n^2+6*n+4)*x-(10*n^2+22*n+8)*x^2+(8*n^2+16*n+4)*x^3)*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[((n^3 + 2*n^2 - n - 2 - (7*n^3 + 16*n^2 + 5*n - 4)*x + (12*n^3 + 18*n^2 - 4*n - 2)*x^2)*P[n - 1, x] - ((2*n^2 + 6*n + 4)*x - (10*n^2 + 22*n + 8)*x^2 + (8*n^2 + 16*n + 4)*x^3)*D[P[n - 1, x], x])/(-n^3 - 3*n^2 - 2*n + (4*n^3 + 2*n^2)*x)]];
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] = ((n^3 + 2*n^2 - n - 2 - (7*n^3 + 16*n^2 + 5*n - 4)*x + (12*n^3 + 18*n^2 - 4*n - 2)*x^2)*v[i - 1] - ((2*n^2 + 6*n + 4)*x - (10*n^2 + 22*n + 8)*x^2 + (8*n^2 + 16*n + 4)*x^3)*deriv(v[i - 1], x))/(-n^3 - 3*n^2 - 2*n + (4*n^3 + 2*n^2)*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.