Real-rooted atlas

A128100 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 18\,x - 8\,x^{2} + n - 17\,n x + 4\,n x^{2} - 4\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(3 + 39\,x + 2\,x^{2} - 24\,x^{3} + 8\,n - 35\,n x - 22\,n x^{2} + 8\,n x^{3} - 12\,n^{2} + 2\,n^{2} x + 8\,n^{2} x^{2} + 3\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(5\,x - 11\,x^{2} - 32\,x^{3} - 16\,x^{4} - 20\,n x - 36\,n x^{2} - 16\,n x^{3} - 5\,n^{2} - 9\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
211
322
4729
516622
63824365
7826812162
8173186544417
9352462180201014
10701109655510052457
11136824841548380305810
122628544840621300165613609
134980116161010440047004231446
1493292420424120114802625252772017
Technical details
Node id
n-7b8a092f0ea30e10 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-27cf973ee2fcb5a7

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:   2;
  4:   7,   2;
  5:  16,   6;
  6:  38,  24,   3;
  7:  82,  68,  12;
  8: 173, 186,  54,  4;
  9: 352, 462, 180, 20;
%F Formula
The row polynomials satisfy (n^3-4*n^2+n+6+(4*n^2-17*n+18)*x+(4*n-8)*x^2)*P(n,x) = (3*n^3-12*n^2+8*n+3+(2*n^3+2*n^2-35*n+39)*x+(8*n^2-22*n+2)*x^2+(8*n-24)*x^3)*P(n-1,x) - (5*n^2+(9*n^2+20*n-5)*x+(4*n^2+36*n+11)*x^2+(16*n+32)*x^3+16*x^4)*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[((3*n^3 - 12*n^2 + 8*n + 3 + (2*n^3 + 2*n^2 - 35*n + 39)*x + (8*n^2 - 22*n + 2)*x^2 + (8*n - 24)*x^3)*P[n - 1, x] - (5*n^2 + (9*n^2 + 20*n - 5)*x + (4*n^2 + 36*n + 11)*x^2 + (16*n + 32)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + n + 6 + (4*n^2 - 17*n + 18)*x + (4*n - 8)*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] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((3*n^3 - 12*n^2 + 8*n + 3 + (2*n^3 + 2*n^2 - 35*n + 39)*x + (8*n^2 - 22*n + 2)*x^2 + (8*n - 24)*x^3)*v[i - 1] - (5*n^2 + (9*n^2 + 20*n - 5)*x + (4*n^2 + 36*n + 11)*x^2 + (16*n + 32)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + n + 6 + (4*n^2 - 17*n + 18)*x + (4*n - 8)*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.