Real-rooted atlas

A100861 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 3\,n\right)\,P_{n}(x) = \left(-6\,n - 27\,n x + 3\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,n - 12\,n x - 18\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
233
399
4365490
5120270390
6450162012153285
716387560850517703
86384378006804034020146244
925056176904408240306180916380
101031408618402551500306180011481757726455
11432300413424014594580224532001262992554244245
121880208204217208532216016839900015155910045467730473049918
138352864101158200483706080113837724013135122005910804903636187074
143826149651329678427875647807764316560114932317508275126860206878171532940579945
Technical details
Node id
n-982f18cab0c327bf · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6d568a62896b03a0

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:     3;
  3:     9;
  4:    36,     54;
  5:   120,    270;
  6:   450,   1620,   1215;
  7:  1638,   7560,   8505;
  8:  6384,  37800,  68040,  34020;
  9: 25056, 176904, 408240, 306180;
%F Formula
The row polynomials satisfy (3*n-6)*P(n,x) = (3*n^2-6*n+(9*n^2-27*n)*x)*P(n-1,x) - (2*n+12*n*x+18*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 6*n + (9*n^2 - 27*n)*x)*P[n - 1, x] - (2*n + 12*n*x + 18*n*x^2)*D[P[n - 1, x], x])/(3*n - 6)]];
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] = 3; for(n = 3, N, my(i = n + 1); v[i] = ((3*n^2 - 6*n + (9*n^2 - 27*n)*x)*v[i - 1] - (2*n + 12*n*x + 18*n*x^2)*deriv(v[i - 1], x))/(3*n - 6)); 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.