Real-rooted atlas

A100861 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(-28 + 36\,x - 8\,x^{2} + 28\,n - 26\,n x + 4\,n x^{2} - 9\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(8\,n - 54\,n x + 46\,n x^{2} - 12\,n x^{3} - 5\,n^{2} + 37\,n^{2} x - 26\,n^{2} x^{2} + 4\,n^{2} x^{3} + n^{3} - 10\,n^{3} x + 4\,n^{3} x^{2} + n^{4} x\right)\,P_{n-1}(x) + \left(-16\,n x^{2} + 20\,n x^{3} - 8\,n x^{4} + 10\,n^{2} x^{2} - 8\,n^{2} x^{3} - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2Pn(1)
00
10
211
333
412618
5403070
610518045330
72318403151386

All 15 rows

Technical details
Node id
n-bb38c4896765a535 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-73cb17ab6470879f

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:   3;
  4:  12,    6;
  5:  40,   30;
  6: 105,  180,    45;
  7: 231,  840,   315;
  8: 448, 2940,  2520,  420;
  9: 792, 8316, 15120, 3780;
%F Formula
The row polynomials satisfy (n^3-9*n^2+28*n-28+(4*n^2-26*n+36)*x+(4*n-8)*x^2)*P(n,x) = (n^3-5*n^2+8*n+(n^4-10*n^3+37*n^2-54*n)*x+(4*n^3-26*n^2+46*n)*x^2+(4*n^2-12*n)*x^3)*P(n-1,x) - ((2*n^3-10*n^2+16*n)*x^2+(8*n^2-20*n)*x^3+8*n*x^4)*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 + 8*n + (n^4 - 10*n^3 + 37*n^2 - 54*n)*x + (4*n^3 - 26*n^2 + 46*n)*x^2 + (4*n^2 - 12*n)*x^3)*P[n - 1, x] - ((2*n^3 - 10*n^2 + 16*n)*x^2 + (8*n^2 - 20*n)*x^3 + 8*n*x^4)*D[P[n - 1, x], x])/(n^3 - 9*n^2 + 28*n - 28 + (4*n^2 - 26*n + 36)*x + (4*n - 8)*x^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 + 8*n + (n^4 - 10*n^3 + 37*n^2 - 54*n)*x + (4*n^3 - 26*n^2 + 46*n)*x^2 + (4*n^2 - 12*n)*x^3)*v[i - 1] - ((2*n^3 - 10*n^2 + 16*n)*x^2 + (8*n^2 - 20*n)*x^3 + 8*n*x^4)*deriv(v[i - 1], x))/(n^3 - 9*n^2 + 28*n - 28 + (4*n^2 - 26*n + 36)*x + (4*n - 8)*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

Same fingerprint ?

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.