Real-rooted atlas

A257613 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
2808016176
39601440608643072
4134402688017600416025662336
52150405376004800001824002700810241443072
63870720116121601317120069888001729280170240409637546496
77741440027095040037330944025589760090639360155366401054208163841084818432

All 15 rows

Technical details
Node id
n-6497b30746d46dc5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-cef044b741d89456

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:           8,            4;
  2:          80,           80,           16;
  3:         960,         1440,          608,           64;
  4:       13440,        26880,        17600,         4160,          256;
  5:      215040,       537600,       480000,       182400,        27008,        1024;
  6:     3870720,     11612160,     13171200,      6988800,      1729280,      170240,        4096;
  7:    77414400,    270950400,    373309440,    255897600,     90639360,    15536640,     1054208,      16384;
  8:  1703116800,   6812467200,  11054776320,   9320693760,   4339722240,  1092833280,   134835200,    6456320,    65536;
  9: 40874803200, 183936614400, 343719936000, 344648908800, 199925268480, 67376332800, 12546355200, 1143244800, 39262208, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+6+(2*n+2)*x)*P(n-1,x) - (2*x+2*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[(2*n + 6 + (2*n + 2)*x)*P[n - 1, x] - (2*x + 2*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] = (2*n + 6 + (2*n + 2)*x)*v[i - 1] - (2*x + 2*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