Real-rooted atlas

A199478 → D

nonnegative coefficients

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

Recurrence

\(\left(n + 8\,n x - 2\,n^{2} + 2\,n^{2} x\right)\,P_{n}(x) = \left(16\,x - 16\,x^{2} - n + 20\,n x + 8\,n x^{2} - 2\,n^{2} - 6\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} - 32\,x^{3} + 2\,n x + 14\,n x^{2} - 16\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
133
2549
371623
49361257
5116460135
61310016832313
715144360192711

All 15 rows

Technical details
Node id
n-36292cabe246c808 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ff141c83922613d6

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:  3;
  2:  5,   4;
  3:  7,  16;
  4:  9,  36,   12;
  5: 11,  64,   60;
  6: 13, 100,  168,   32;
  7: 15, 144,  360,  192;
  8: 17, 196,  660,  640,  80;
  9: 19, 256, 1092, 1600, 560;
%F Formula
The row polynomials satisfy -(2*n^2-n-(2*n^2+8*n)*x)*P(n,x) = -(2*n^2+n+(6*n^2-20*n-16)*x-(8*n^2+8*n-16)*x^2)*P(n-1,x) + ((2*n+1)*x+(14*n+4)*x^2-(16*n+32)*x^3)*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[(-(2*n^2 + n + (6*n^2 - 20*n - 16)*x - (8*n^2 + 8*n - 16)*x^2)*P[n - 1, x] + ((2*n + 1)*x + (14*n + 4)*x^2 - (16*n + 32)*x^3)*D[P[n - 1, x], x])/(-2*n^2 + n + (2*n^2 + 8*n)*x)]];
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] = (-(2*n^2 + n + (6*n^2 - 20*n - 16)*x - (8*n^2 + 8*n - 16)*x^2)*v[i - 1] + ((2*n + 1)*x + (14*n + 4)*x^2 - (16*n + 32)*x^3)*deriv(v[i - 1], x))/(-2*n^2 + n + (2*n^2 + 8*n)*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.