Real-rooted atlas

A119462 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 14\,x - 4\,x^{2} + 7\,n - 18\,n x + 4\,n x^{2} - 4\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-2 + 16\,x + 6\,x^{2} - 4\,x^{3} + 3\,n - 17\,n x - 10\,n x^{2} + 4\,n x^{3} - 2\,n^{2} + 4\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-4\,x - 8\,x^{2} + 20\,x^{3} - 8\,x^{4} - 2\,n x + 10\,n x^{2} - 8\,n x^{3} + 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3Pn(1)
011
122
2426
38614
41416232
522401072
63290362160
74418211214352

All 15 rows

Technical details
Node id
n-b4d8a8c1e41e05a6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2cceb230b30e91a5

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:  2;
  2:  4,   2;
  3:  8,   6;
  4: 14,  16,   2;
  5: 22,  40,  10;
  6: 32,  90,  36,   2;
  7: 44, 182, 112,  14;
  8: 58, 336, 308,  64,  2;
  9: 74, 576, 756, 240, 18;
%F Formula
The row polynomials satisfy (n^3-4*n^2+7*n-4+(4*n^2-18*n+14)*x+(4*n-4)*x^2)*P(n,x) = (n^3-2*n^2+3*n-2+(n^3-17*n+16)*x+(4*n^2-10*n+6)*x^2+(4*n-4)*x^3)*P(n-1,x) + ((2*n^2-2*n-4)*x-(2*n^2-10*n+8)*x^2-(8*n-20)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 2*n^2 + 3*n - 2 + (n^3 - 17*n + 16)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + ((2*n^2 - 2*n - 4)*x - (2*n^2 - 10*n + 8)*x^2 - (8*n - 20)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + 7*n - 4 + (4*n^2 - 18*n + 14)*x + (4*n - 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; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - 2*n^2 + 3*n - 2 + (n^3 - 17*n + 16)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*v[i - 1] + ((2*n^2 - 2*n - 4)*x - (2*n^2 - 10*n + 8)*x^2 - (8*n - 20)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + 7*n - 4 + (4*n^2 - 18*n + 14)*x + (4*n - 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

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.