Real-rooted atlas

A173018 → reverse (shift +1) → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 - 6\,x + 6\,x^{2} - 2\,n + 8\,n x - 11\,n x^{2} - 2\,n^{2} x + 6\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(x - 3\,x^{2} + 2\,x^{3} - n x + 4\,n x^{2} - 3\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-5 + 8\,x + 3\,n - 8\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
211
3426
41122336
526132784240
65760490622851800
7120238272484764600615120
8247858646857624762146514827141120
95022921626470262476044117087648351481451520
1010139568013655765241416655177027311523348808104916329600
11203630527466104643895245678621240584286841542441612210961832410199584000
1240839565423056305526527389681256143097507371646422931881501480430443940830112634508800
1381782959452136600350169312614075281075401365103202410539350556338625228040980105014797260899581237362124800
1416369907462859523235810286968700642163143151678270565521957982326441027461029042314567957519841078604991045419642813566658892800
Technical details
Node id
n-b59d7fbabc61ed1c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b00b790aa67df404

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
  • 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:   0;
  2:   1;
  3:   4,     2;
  4:  11,    22,      3;
  5:  26,   132,     78,      4;
  6:  57,   604,    906,    228,      5;
  7: 120,  2382,   7248,   4764,    600,     6;
  8: 247,  8586,  46857,  62476,  21465,  1482,    7;
  9: 502, 29216, 264702, 624760, 441170, 87648, 3514, 8;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(2*n-2+(2*n^2-8*n+6)*x+(n^3-6*n^2+11*n-6)*x^2)*P(n-2,x) - ((n-1)*x+(n^2-4*n+3)*x^2-(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (3*n-5+(2*n^2-8*n+8)*x)*P(n-1,x) + ((n-2)*x-(n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 + (2*n^2 - 8*n + 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*P[n - 2, x] - ((n - 1)*x + (n^2 - 4*n + 3)*x^2 - (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (3*n - 5 + (2*n^2 - 8*n + 8)*x)*P[n - 1, x] + ((n - 2)*x - (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 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] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(2*n - 2 + (2*n^2 - 8*n + 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*v[i - 2] - ((n - 1)*x + (n^2 - 4*n + 3)*x^2 - (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (3*n - 5 + (2*n^2 - 8*n + 8)*x)*v[i - 1] + ((n - 2)*x - (n - 2)*x^2)*deriv(v[i - 1], x))/(n - 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.