Real-rooted atlas

A100862 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,x^{2} + 2\,n - n x - 2\,n x^{2} + n^{2} x\right)\,P_{n}(x) = \left(n - 5\,n x - 9\,n x^{2} - 2\,n x^{3} + 4\,n^{2} + 8\,n^{2} x + 3\,n^{2} x^{2} - n^{3} - n^{3} x\right)\,P_{n-1}(x) + \left(-2\,n - 8\,n x - 10\,n x^{2} - 4\,n x^{3} + 2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2527
32118342
486114424246
53556403758051455
614973420277596013568793
76475179341869091002100210754516

All 15 rows

Technical details
Node id
n-9ecc547d6c6674d2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-fe89f9278b568edf

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:      5,      2;
  3:     21,     18,      3;
  4:     86,    114,     42,      4;
  5:    355,    640,    375,     80,      5;
  6:   1497,   3420,   2775,    960,    135,     6;
  7:   6475,  17934,  18690,   9100,   2100,   210,    7;
  8:  28772,  93716, 119616,  75740,  24920,  4116,  308,   8;
  9: 131337, 492048, 744534, 583128, 252315, 59976, 7434, 432, 9;
%F Formula
The row polynomials satisfy (2*n-2+(n^2-n)*x-(2*n-2)*x^2)*P(n,x) = -(n^3-4*n^2-n+(n^3-8*n^2+5*n)*x-(3*n^2-9*n)*x^2+2*n*x^3)*P(n-1,x) + (2*n^2-2*n+(4*n^2-8*n)*x+(2*n^2-10*n)*x^2-4*n*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[(-(n^3 - 4*n^2 - n + (n^3 - 8*n^2 + 5*n)*x - (3*n^2 - 9*n)*x^2 + 2*n*x^3)*P[n - 1, x] + (2*n^2 - 2*n + (4*n^2 - 8*n)*x + (2*n^2 - 10*n)*x^2 - 4*n*x^3)*D[P[n - 1, x], x])/(2*n - 2 + (n^2 - n)*x - (2*n - 2)*x^2)]];
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] = (-(n^3 - 4*n^2 - n + (n^3 - 8*n^2 + 5*n)*x - (3*n^2 - 9*n)*x^2 + 2*n*x^3)*v[i - 1] + (2*n^2 - 2*n + (4*n^2 - 8*n)*x + (2*n^2 - 10*n)*x^2 - 4*n*x^3)*deriv(v[i - 1], x))/(2*n - 2 + (n^2 - n)*x - (2*n - 2)*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.