Real-rooted atlas

A034867 → D+1

nonnegative coefficients

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

Recurrence

\(\left(8\,n - 14\,n x + 4\,n x^{2} - 3\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(6 - 16\,x + 14\,x^{2} - 4\,x^{3} + 5\,n - 5\,n x - 10\,n x^{2} + 4\,n x^{3} + 4\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(4\,x - 16\,x^{2} + 20\,x^{3} - 8\,x^{4} - 2\,n x + 10\,n x^{2} - 8\,n x^{3} + 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
2415
38412
41512128
52632664
64277241144
764168808320
893336234401704
9130624612160101536
10176108914525506013328
11232180431681672280127168
12299286064354576110584115360
133784368122981144038224481432768
1447064612230826455118301995112169632
Technical details
Node id
n-24485a3583f83408 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-efd07601c6f432c5

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,   1;
  3:   8,   4;
  4:  15,  12,   1;
  5:  26,  32,   6;
  6:  42,  77,  24,   1;
  7:  64, 168,  80,   8;
  8:  93, 336, 234,  40,  1;
  9: 130, 624, 612, 160, 10;
%F Formula
The row polynomials satisfy (n^3-3*n^2+8*n+(4*n^2-14*n)*x+4*n*x^2)*P(n,x) = (n^3+5*n+6+(n^3-5*n-16)*x+(4*n^2-10*n+14)*x^2+(4*n-4)*x^3)*P(n-1,x) + ((2*n^2-2*n+4)*x-(2*n^2-10*n+16)*x^2-(8*n-20)*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 + 5*n + 6 + (n^3 - 5*n - 16)*x + (4*n^2 - 10*n + 14)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + ((2*n^2 - 2*n + 4)*x - (2*n^2 - 10*n + 16)*x^2 - (8*n - 20)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 8*n + (4*n^2 - 14*n)*x + 4*n*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 + 5*n + 6 + (n^3 - 5*n - 16)*x + (4*n^2 - 10*n + 14)*x^2 + (4*n - 4)*x^3)*v[i - 1] + ((2*n^2 - 2*n + 4)*x - (2*n^2 - 10*n + 16)*x^2 - (8*n - 20)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 8*n + (4*n^2 - 14*n)*x + 4*n*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.