Real-rooted atlas

A257624 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + 2\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110515
213013025285
32080312012901256615
439520790405109011570625181845
5869440217360019240007124009881031255781375
621736000652080007350720038334400912093082173015625208743885
76086080002130128000292131840019779760006826164801110127206730090781258438467815

All 15 rows

Technical details
Node id
n-074080256a21a584 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1d32ce7fdc01b2c4

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:           10,             5;
  2:          130,           130,            25;
  3:         2080,          3120,          1290,           125;
  4:        39520,         79040,         51090,         11570,           625;
  5:       869440,       2173600,       1924000,        712400,         98810,          3125;
  6:     21736000,      65208000,      73507200,      38334400,       9120930,        821730,        15625;
  7:    608608000,    2130128000,    2921318400,    1977976000,     682616480,     111012720,      6730090,       78125;
  8:  18866848000,   75467392000,  122025904000,  101941840000,   46595305120,   11332834240,   1308631090,    54621970,    390625;
  9: 641472832000, 2886627744000, 5378877504000, 5355141792000, 3063933512640, 1007444037600, 179597776800, 15105027600, 440882010, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+7+(3*n+2)*x)*P(n-1,x) - (3*x+3*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[(3*n + 7 + (3*n + 2)*x)*P[n - 1, x] - (3*x + 3*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] = (3*n + 7 + (3*n + 2)*x)*v[i - 1] - (3*x + 3*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