Real-rooted atlas

A142460 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
21414129
3168252861507
4285657123374518112461
5628321570801318804074031101395643
6169646450893925540640259896046991018662115414029
7542868481900039682542182721605357604722228052996441119741711678747

All 15 rows

Technical details
Node id
n-59a56c7d8834e636 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b94fba28f209154b

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:           1;
  1:           2,            1;
  2:          14,           14,            1;
  3:         168,          252,           86,            1;
  4:        2856,         5712,         3374,          518,            1;
  5:       62832,       157080,       131880,        40740,         3110,            1;
  6:     1696464,      5089392,      5540640,      2598960,       469910,        18662,           1;
  7:    54286848,    190003968,    254218272,    160535760,     47222280,      5299644,      111974,         1;
  8:  2008613376,   8034453504,  12754016352,  10141461792,   4174029720,    819152208,    59079902,    671846,       1;
  9: 84361761792, 379627928064, 697368849408, 669193975296, 355506660432, 101580211656, 13815394152, 654581844, 4031078, 1;
%F Formula
The row polynomials satisfy P(n,x) = (5*n-3+(5*n-4)*x)*P(n-1,x) - (5*x+5*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5*n - 3 + (5*n - 4)*x)*P[n - 1, x] - (5*x + 5*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5*n - 3 + (5*n - 4)*x)*v[i - 1] - (5*x + 5*x^2)*deriv(v[i - 1], x)); 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