Real-rooted atlas

A008292 → γ (scale 1/2) (shift +1) → D → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 + 8\,x + 8\,x^{2} - 6\,n - 18\,n x - 14\,n x^{2} + 5\,n^{2} + 12\,n^{2} x + 7\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 + 5\,x + 8\,x^{2} + 4\,x^{3} - 2\,n - 9\,n x - 13\,n x^{2} - 6\,n x^{3} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(7 + 12\,x - 7\,n - 10\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2 + 6\,x + 4\,x^{2} - n - 3\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
211
344
419827
59468162
65195641021185
73144451214889144
8209033748818816198479191
915141832566821897644224740286
10118894729845722528814787456552807545069
1110064924288883842955640812478432174752082735668
1291426347295386120354561960190052640415515602097024975075651
138872964223187105860439272336028603420808566749608923862412273381306
149164847535362319214445640851967043292735680165742699202727075360104111728164503481337
Technical details
Node id
n-85d3edd135bd226f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-20bf382ca2f671b0

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half 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;
  4:     19,      8;
  5:     94,     68;
  6:    519,    564,    102;
  7:   3144,   4512,   1488;
  8:  20903,  37488,  18816,  1984;
  9: 151418, 325668, 218976, 44224;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(n^3-5*n^2+6*n-2+(2*n^3-12*n^2+18*n-8)*x+(n^3-7*n^2+14*n-8)*x^2)*P(n-2,x) + (n^2-2*n+1+(4*n^2-9*n+5)*x+(5*n^2-13*n+8)*x^2+(2*n^2-6*n+4)*x^3)*d/dx P(n-2,x) + (2*n^2-7*n+7+(2*n^2-10*n+12)*x)*P(n-1,x) - (n-2+(3*n-6)*x+(2*n-4)*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[(-(n^3 - 5*n^2 + 6*n - 2 + (2*n^3 - 12*n^2 + 18*n - 8)*x + (n^3 - 7*n^2 + 14*n - 8)*x^2)*P[n - 2, x] + (n^2 - 2*n + 1 + (4*n^2 - 9*n + 5)*x + (5*n^2 - 13*n + 8)*x^2 + (2*n^2 - 6*n + 4)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 7*n + 7 + (2*n^2 - 10*n + 12)*x)*P[n - 1, x] - (n - 2 + (3*n - 6)*x + (2*n - 4)*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] = (-(n^3 - 5*n^2 + 6*n - 2 + (2*n^3 - 12*n^2 + 18*n - 8)*x + (n^3 - 7*n^2 + 14*n - 8)*x^2)*v[i - 2] + (n^2 - 2*n + 1 + (4*n^2 - 9*n + 5)*x + (5*n^2 - 13*n + 8)*x^2 + (2*n^2 - 6*n + 4)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 7*n + 7 + (2*n^2 - 10*n + 12)*x)*v[i - 1] - (n - 2 + (3*n - 6)*x + (2*n - 4)*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

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.