Real-rooted atlas

A100861 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 2\,x + 4\,x^{2} + n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(-4 - 4\,x - 2\,x^{2} - 4\,x^{3} + 2\,n + n x + 2\,n x^{2} + 4\,n x^{3} + n^{2} + 4\,n^{2} x + 4\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-8 - 28\,x - 40\,x^{2} - 28\,x^{3} - 8\,x^{4} - 6\,n - 20\,n x - 22\,n x^{2} - 8\,n x^{3} - 2\,n^{2} - 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2314
37310
42218343
5667015151
622633013515706
777813867351053004

All 15 rows

Technical details
Node id
n-822dfdb87dc2a6f3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-21eb1a366779e332

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:     1;
  2:     3,     1;
  3:     7,     3;
  4:    22,    18,     3;
  5:    66,    70,    15;
  6:   226,   330,   135,   15;
  7:   778,  1386,   735,  105;
  8:  2892,  6328,  4620, 1260, 105;
  9: 10972, 28008, 24948, 8820, 945;
%F Formula
The row polynomials satisfy (n^2+n-2+(4*n-2)*x+4*x^2)*P(n,x) = (n^3+n^2+2*n-4+(n^3+4*n^2+n-4)*x+(4*n^2+2*n-2)*x^2+(4*n-4)*x^3)*P(n-1,x) - (2*n^2+6*n+8+(4*n^2+20*n+28)*x+(2*n^2+22*n+40)*x^2+(8*n+28)*x^3+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[((n^3 + n^2 + 2*n - 4 + (n^3 + 4*n^2 + n - 4)*x + (4*n^2 + 2*n - 2)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] - (2*n^2 + 6*n + 8 + (4*n^2 + 20*n + 28)*x + (2*n^2 + 22*n + 40)*x^2 + (8*n + 28)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(n^2 + n - 2 + (4*n - 2)*x + 4*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] = ((n^3 + n^2 + 2*n - 4 + (n^3 + 4*n^2 + n - 4)*x + (4*n^2 + 2*n - 2)*x^2 + (4*n - 4)*x^3)*v[i - 1] - (2*n^2 + 6*n + 8 + (4*n^2 + 20*n + 28)*x + (2*n^2 + 22*n + 40)*x^2 + (8*n + 28)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(n^2 + n - 2 + (4*n - 2)*x + 4*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. [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.