Real-rooted atlas

type_b_eulerian → γ (scale 1/4) → D → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(3 + 8\,x + 8\,x^{2} - 7\,n - 18\,n x - 14\,n x^{2} + 5\,n^{2} + 12\,n^{2} x + 7\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 6\,x^{2} + 4\,x^{3} - 4\,n x - 10\,n x^{2} - 6\,n x^{3} + 2\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(5 + 12\,x - 6\,n - 10\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x + 4\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2Pn(1)
00
10
211
355
4281038
5180122302
6132013241832827
71092014532415529607

All 15 rows

Technical details
Node id
n-2835967399023614 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1644966885f7161f

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
  • 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:       0;
  1:       0;
  2:       1;
  3:       5;
  4:      28,      10;
  5:     180,     122;
  6:    1320,    1324,     183;
  7:   10920,   14532,    4155;
  8:  100800,  167328,   73704,   5540;
  9: 1028160, 2046240, 1225080, 202084;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(n^3-5*n^2+7*n-3+(2*n^3-12*n^2+18*n-8)*x+(n^3-7*n^2+14*n-8)*x^2)*P(n-2,x) + ((2*n^2-4*n+2)*x+(4*n^2-10*n+6)*x^2+(2*n^2-6*n+4)*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+5+(2*n^2-10*n+12)*x)*P(n-1,x) - ((2*n-4)*x+(2*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 5*n^2 + 7*n - 3 + (2*n^3 - 12*n^2 + 18*n - 8)*x + (n^3 - 7*n^2 + 14*n - 8)*x^2)*P[n - 2, x] + ((2*n^2 - 4*n + 2)*x + (4*n^2 - 10*n + 6)*x^2 + (2*n^2 - 6*n + 4)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + 5 + (2*n^2 - 10*n + 12)*x)*P[n - 1, x] - ((2*n - 4)*x + (2*n - 4)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(n^3 - 5*n^2 + 7*n - 3 + (2*n^3 - 12*n^2 + 18*n - 8)*x + (n^3 - 7*n^2 + 14*n - 8)*x^2)*v[i - 2] + ((2*n^2 - 4*n + 2)*x + (4*n^2 - 10*n + 6)*x^2 + (2*n^2 - 6*n + 4)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + 5 + (2*n^2 - 10*n + 12)*x)*v[i - 1] - ((2*n - 4)*x + (2*n - 4)*x^2)*deriv(v[i - 1], x))/(n - 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.