Real-rooted atlas

A107131 → D

nonnegative coefficients

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

Recurrence

\(\left(2 - 2\,x - n + n x - n^{2} + n^{2} x\right)\,P_{n}(x) = \left(12 + 2\,x - 2\,x^{2} - 16\,n + 11\,n x - n x^{2} + 4\,n^{2} - 7\,n^{2} x + 3\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(8\,x + 2\,x^{2} - x^{3} - 8\,n x + 10\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2123
30639
40418426
5003040575
60015120756216
70001403501267623

All 15 rows

Technical details
Node id
n-7f97bfe18fd4cd4e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-208c74c0c32ded3e

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 0;
  1: 1;
  2: 1, 2;
  3: 0, 6,  3;
  4: 0, 4, 18,   4;
  5: 0, 0, 30,  40,   5;
  6: 0, 0, 15, 120,  75,    6;
  7: 0, 0,  0, 140, 350,  126,    7;
  8: 0, 0,  0,  56, 700,  840,  196,   8;
  9: 0, 0,  0,   0, 630, 2520, 1764, 288, 9;
%F Formula
The row polynomials satisfy -(n^2+n-2-(n^2+n-2)*x)*P(n,x) = (4*n^2-16*n+12-(7*n^2-11*n-2)*x+(3*n^2-n-2)*x^2)*P(n-1,x) - ((8*n-8)*x-(10*n+2)*x^2+(2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 16*n + 12 - (7*n^2 - 11*n - 2)*x + (3*n^2 - n - 2)*x^2)*P[n - 1, x] - ((8*n - 8)*x - (10*n + 2)*x^2 + (2*n + 1)*x^3)*D[P[n - 1, x], x])/(-n^2 - n + 2 + (n^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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 16*n + 12 - (7*n^2 - 11*n - 2)*x + (3*n^2 - n - 2)*x^2)*v[i - 1] - ((8*n - 8)*x - (10*n + 2)*x^2 + (2*n + 1)*x^3)*deriv(v[i - 1], x))/(-n^2 - n + 2 + (n^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. [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.