Real-rooted atlas

A269951 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-20 + 9\,n - n^{2}\right)\,P_{n-4}(x) + \left(20 - 8\,x - 13\,n + 2\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(3 + 4\,x - x^{2} + 2\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-3 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2\,x\)
\(P_{3}(x) = 4\,x + 3\,x^{2}\)
\(P_{4}(x) = 10\,x + 15\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2022
30437
401015429
503269365142
60130348260706814
706522007188072512075391
803914132871453670751680189840689

All 15 rows

Technical details
Node id
n-3503b708139a235b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-110d29bcfc73cc03

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,      4,      3;
   4: 0,     10,     15,       4;
   5: 0,     32,     69,      36,      5;
   6: 0,    130,    348,     260,     70,      6;
   7: 0,    652,   2007,    1880,    725,    120,     7;
   8: 0,   3914,  13287,   14536,   7075,   1680,   189,    8;
   9: 0,  27400, 100125,  122724,  70420,  21210,  3430,  280,   9;
  10: 0, 219202, 849216, 1136264, 737660, 263634, 54292, 6384, 396, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-9*n+20)*P(n-4,x) + (2*n^2-13*n+20+(2*n-8)*x)*P(n-3,x) - (n^2-2*n-3+(2*n-4)*x+x^2)*P(n-2,x) + (2*n-3+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2*x, P(3,x) = 4*x + 3*x^2, P(4,x) = 10*x + 15*x^2 + 4*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2*x;
P[3, x_] := 4*x + 3*x^2;
P[4, x_] := 10*x + 15*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 9*n + 20)*P[n - 4, x] + (2*n^2 - 13*n + 20 + (2*n - 8)*x)*P[n - 3, x] - (n^2 - 2*n - 3 + (2*n - 4)*x + x^2)*P[n - 2, x] + (2*n - 3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; v[3] = 4*x + 3*x^2; v[4] = 10*x + 15*x^2 + 4*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 9*n + 20)*v[i - 4] + (2*n^2 - 13*n + 20 + (2*n - 8)*x)*v[i - 3] - (n^2 - 2*n - 3 + (2*n - 4)*x + x^2)*v[i - 2] + (2*n - 3 + 2*x)*v[i - 1]); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.