Real-rooted atlas

A129710 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2} - 2\,n^{2} + 4\,n^{2} x + 3\,n^{2} x^{2} - 2\,n^{3} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(-10\,x^{2} - 4\,x^{3} + 2\,n + 18\,n x + 17\,n x^{2} - 2\,n x^{3} - 2\,n^{2} - 4\,n^{2} x + 8\,n^{2} x^{2} + 4\,n^{2} x^{3} - 2\,n^{3} x - 3\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 10\,x^{3} - 8\,x^{4} + 2\,n x + 10\,n x^{2} + 5\,n x^{3} - 12\,n x^{4} + 2\,n^{2} x^{2} + 7\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
2224
3268
4210315
52141228
621827451
7222482092
822675565164
923010812030290
102341472201006509
1123819236425042888
1224224356052516271541
13246300816980462562662
1425036311401680109224584580
Technical details
Node id
n-530cbe1f55af9e87 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-da2c2ef4620dc58e

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: 2;
  2: 2,  2;
  3: 2,  6;
  4: 2, 10,   3;
  5: 2, 14,  12;
  6: 2, 18,  27,   4;
  7: 2, 22,  48,  20;
  8: 2, 26,  75,  56,  5;
  9: 2, 30, 108, 120, 30;
%F Formula
The row polynomials satisfy -(2*n^2-2*n+(2*n^3-4*n^2-4*n)*x-(n^3+3*n^2+2*n)*x^2)*P(n,x) = -(2*n^2-2*n+(2*n^3+4*n^2-18*n)*x+(3*n^3-8*n^2-17*n+10)*x^2-(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) + (2*n*x+(2*n^2+10*n-2)*x^2+(7*n^2+5*n-10)*x^3-(4*n^2+12*n+8)*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[(-(2*n^2 - 2*n + (2*n^3 + 4*n^2 - 18*n)*x + (3*n^3 - 8*n^2 - 17*n + 10)*x^2 - (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] + (2*n*x + (2*n^2 + 10*n - 2)*x^2 + (7*n^2 + 5*n - 10)*x^3 - (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(-2*n^2 + 2*n - (2*n^3 - 4*n^2 - 4*n)*x + (n^3 + 3*n^2 + 2*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^2 - 2*n + (2*n^3 + 4*n^2 - 18*n)*x + (3*n^3 - 8*n^2 - 17*n + 10)*x^2 - (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] + (2*n*x + (2*n^2 + 10*n - 2)*x^2 + (7*n^2 + 5*n - 10)*x^3 - (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(-2*n^2 + 2*n - (2*n^3 - 4*n^2 - 4*n)*x + (n^3 + 3*n^2 + 2*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

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.